0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
Since $(G,\mathscr{G})$ has strongly induced copies, the same applies to
$(\Pi_{N},\mathscr{P}_{N})$ (see Lemma 3.4 and its proof). So altogether, when
viewed as an ordered $f$-partite structure the last picture is as required by
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.
∎

###  12.4. Girth resurrection

Now we shall finally complete the proof of Proposition 10.14. So we consider
an arbitrary nonempty nondecreasing sequence
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}^{\times}_{\leq}$ and assume
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.
Let $\varUpsilon$ be the construction provided by Lemma 11.10. Evidently
$\varUpsilon$ is a Ramsey construction for
$((2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-trains. So in view of Lemma 12.6 it only remains to exhibit
an $((2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-amenable partite lemma.

By restricting our attention to unordered $k$-partite $k$-uniform trains we
can regard $\varUpsilon$ as a partite lemma for trains and in this manner
$\Xi=\mathrm{PC}(\varUpsilon,\varUpsilon)$ becomes a partite lemma as well. It
will turn out that $\Xi$ has the required amenability property.

######  Lemma 12.7.

For every $k$-partite $k$-uniform train
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ of
height $m+1$ with
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>(2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and every number of colours $r$ the train system
$\Xi_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ is defined. Moreover, the train
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
has the same parameter as $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$,
the copies in $\mathscr{H}$ are strongly induced, their intersections are
clean, and $\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.

###### Proof.

Owing to Lemma 11.10 the $k$-partite $k$-uniform train system
$\varUpsilon_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}},\mathscr{G})$ satisfies
$\mathscr{G}\longrightarrow(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})_{r}$, its copies are strongly induced, and the parameters of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
and $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle G\hfil$\crcr}}}$
are the same. Without loss of generality we can assume that every edge of $G$
belongs to some copy in $\mathscr{G}$.

Let $E(G)=\\{e(1),\dots,e(N)\\}$ enumerate the edges of $G$. We will create a
sequence $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha},\mathscr{P}_{\alpha},\psi_{\alpha})_{\alpha\leq
N}$ of train pictures over $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}},\mathscr{G})$ by means of the partite construction method.
Clearly, picture zero $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{0},\mathscr{P}_{0},\psi_{0})$
exists and it has strongly induced copies with clean intersections. Moreover,
Fact 11.9 yields
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{0},\mathscr{P}_{0}^{+})>(2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$,
which implies, in particular,
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{0},\mathscr{P}_{0}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.

Now suppose inductively that for some positive $\alpha\leq N$ we have just
constructed a train picture $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1},\mathscr{P}_{\alpha-1},\psi_{\alpha-1})$
such that the copies in $\mathscr{P}_{\alpha-1}$ are strongly induced, their
intersections are clean, and
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1},\mathscr{P}_{\alpha-1}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.

Combined with Lemma 8.17 and the linearity of $\Pi_{\alpha-1}$ this shows, in
particular, that
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1}^{e(\alpha)})>(2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and thus Lemma 11.10 yields a $k$-partite $k$-uniform train system

$\varUpsilon_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1}^{e(\alpha)})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha},\mathscr{H}_{\alpha})$

with the correct parameter satisfying
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha},\mathscr{H}_{\alpha})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
Due to Lemma 12.3 the next train picture

$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha},\mathscr{P}_{\alpha},\psi_{\alpha})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1},\mathscr{P}_{\alpha-1},\psi_{\alpha-1})\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr+\cr\times\crcr}}}}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha},\mathscr{H}_{\alpha})$

exists. By Lemma 3.4 the copies in $\mathscr{P}_{\alpha}$ are again strongly
induced and their intersections are clean. Moreover, Lemma 12.4 tells us
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha},\mathscr{P}_{\alpha}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
Thus the construction goes on and the final train picture we eventually reach
has the desired properties. ∎

By Lemma 4.13 the systems $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ produced by $\Xi$ satisfy
$\mathrm{Girth}(H,\mathscr{H})>(2,2)$ and, consequently, $\Xi$ is indeed
$((2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-amenable. Thus the discussion at the beginning of this
subsection shows that the construction $\mathrm{PC}(\varUpsilon,\Xi)$
exemplifies
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.
This concludes the proof of Proposition 10.14. For later reference we
formulate a straightforward consequence of the results we currently have.

######  Corollary 12.8.

For every $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}^{\times}$ there exists a Ramsey construction
for $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}$-trains.

###### Proof.

Due to Lemma 10.13 and Proposition 10.14 we have
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(2)^{m}}$ for every positive integer
$m$. Clearly, every construction exemplifying
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(2)^{m}}$ for
$m=|\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}|$
is a Ramsey construction for $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}$-trains. ∎

###  12.5. Revisability

The remainder of this section is devoted to the proof of Proposition 10.15. To
this end we fix a (possibly empty) nondecreasing sequence
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$ as
well as an integer $g$ such that
$\inf(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})>g\geq 2$. Let $\ell$ be the length of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and write $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}=(g_{1},\dots,g_{\ell})$. We assume that for every positive
integer $m$ there exists a construction $\Psi^{m}$ exemplifying the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.
Our deduction of
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
consists of three steps.

  1. $\bullet$

First, we explain how the constructions $\Psi^{m}$ are going to be used.

  2. $\bullet$

Second, we perform a “diagonal” partite construction in order to derive a
$((g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-amenable partite lemma.

  3. $\bullet$

Third, we conclude
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.

The given constructions $\Psi^{m}$ will only be used as partite lemmata, i.e.,
we shall only apply them to unordered $k$-partite $k$-uniform trains. More
precisely, each $\Psi^{m}$ will be reinterpreted as a partite lemma applicable
to certain trains $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ of
height $\ell+1$ that will be called $m$-revisable. Here is the definition of
this concept.

######  Definition 12.9.

A $k$-partite $k$-uniform train
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0},\dots,\equiv_{\ell+1})$ of height $\ell+1$ with
index set $I$ and parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
A\hfil$\crcr}}}=(A_{1},\dots,A_{\ell+1})$ is said to be $m$-revisable for a
positive integer $m$ if there exist equivalence relations
$\sim_{1},\dots,\sim_{m-1}$ on $E(F)$ and an $m$-tuple
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}=(B_{1},\dots,B_{m})\in\powerset(A_{1})^{m}$ such that

  1. $\bullet$

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet}=(F,\equiv_{0},\sim_{1},\dots,\sim_{m-1},\equiv_{1},\ldots,\equiv_{\ell+1})$
is a train of height $\ell+m$ with parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}\circ(A_{2},\dots,A_{\ell+1})$;

  2. $\bullet$

$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet})>(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$;

  3. $\bullet$

and $|B_{\mu}|\leq 1$ for every $\mu\in[m]$.

In this situation we call the pair
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet},\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle B\hfil$\crcr}}})$
an $m$-revision of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.

We illustrate this notion with an easy example that will later be applied to
the constituents of a train picture zero.

######  Fact 12.10.

Let $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ be
a $k$-partite $k$-uniform train of height $\ell+1$. If its underlying
hypergraph is a matching, possibly together with some isolated vertices, then
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
$1$-revisable.

###### Proof.

For every $\lambda\in[\ell+1]$ any two distinct $(\lambda-1)$-wagons of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
are vertex-disjoint. Thus we have
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>(g)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}},(\varnothing))$ is a $1$-revision of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$. ∎

Here is a necessary condition for revisability that could be shown to be quite
far from being sufficient.

######  Fact 12.11.

If a $k$-partite $k$-uniform train of height $\ell+1$ is $m$-revisable for
some positive integer $m$, then
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.

###### Proof.

Let $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$,
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle B\hfil$\crcr}}}$,
and $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet}$ be as in Definition 12.9. We need to prove that for
every $\lambda\in[\ell+1]$ and every $\lambda$-wagon $W$ of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
the $(\lambda-1)$-wagons in $W$ form a set system whose girth exceeds the
$\lambda^{\mathrm{th}}$ entry of
$(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
For $\lambda\neq 1$ this follows immediately from
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet})>(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.

Thus it remains to show $\mathrm{girth}(W)>g+1$ for every $1$-wagon $W$ of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.
To this end we notice that by restricting the first $m+1$ equivalence
relations of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet}$ to $E(W)$ we obtain a train
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
W\hfil$\crcr}}}=(W,\equiv_{0}^{W},\sim_{1}^{W},\dots,\sim_{m-1}^{W},\equiv^{W}_{1})$
of height $m$. We know
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
W\hfil$\crcr}}})>(g)^{m}$ and the parameter of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle W\hfil$\crcr}}}$ is
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}=(B_{1},\dots,B_{m})$. As we required $|B_{\mu}|\leq 1$ for
every $\mu\in[m]$, all assumptions of Lemma 10.9 are satisfied and we have
indeed $\mathrm{girth}(W)>g+1$. ∎

Given an $m$-revisable $k$-partite $k$-uniform train
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ as
well as a number of colours $r$ we can take an arbitrary $m$-revision
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet},\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle B\hfil$\crcr}}})$
of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
and construct the train system

$\Psi^{m}_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet})=(H,\equiv_{0}^{H},\sim_{1}^{H},\dots,\sim_{m-1}^{H},\equiv_{1}^{H},\ldots,\equiv_{\ell+1}^{H},\mathscr{H}_{\bullet})\,.$
(12.4)

The outcome does not depend on
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
alone, for there could be many distinct $m$-revisions of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.
This ambiguity, however, is irrelevant to the main concern of this subsection
and it will be convenient to write

$\Psi^{m}_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})\,,$

where $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv_{0}^{H},\ldots,\equiv_{\ell+1}^{H})$ and the copies
in $\mathscr{H}\subseteq\binom{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}}{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}}$
are obtained from the copies of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet}$ in $\mathscr{H}_{\bullet}$ by forgetting the
$\sim$-relations.

######  Lemma 12.12.

If $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
denotes an $m$-revisable $k$-partite $k$-uniform train of height $\ell+1$ with
parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$,
and $r$ is a number of colours, then the train system
$\Psi^{m}_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ has again the parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
and satisfies

$\mathscr{H}\longrightarrow(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})_{r}\,,\quad\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}})>(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}},\quad\text{ and
}\quad\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})>(g)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}\,.$

###### Proof.

We keep using the notation from Definition 12.9 and (12.4). As a first step we
shall show that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
parametrises $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$.
Due to
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
the train

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\bullet}=(H,\equiv_{0}^{H},\sim_{1}^{H},\dots,\sim_{m-1}^{H},\equiv_{1}^{H},\ldots,\equiv_{\ell+1}^{H})$

has the same parameter as $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}_{\bullet}$, i.e.,
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}\circ(A_{2},\dots,A_{\ell+1})$. Setting for simplicity
$\sim^{H}_{0}=\equiv^{H}_{0}$ and $\sim^{H}_{m}=\equiv^{H}_{1}$ this means
that for any two edges $e,e^{\prime}\in E(H)$ the following two statements
hold.

  1. (0 )

If $\mu\in[m]$, $e\sim^{H}_{\mu}e^{\prime}$, and
$e\not\sim^{H}_{\mu-1}e^{\prime}$, then $e\cap e^{\prime}\subseteq
V_{B_{\mu}}(H)$.

  2. (0 )

If $\lambda\in[2,\ell+1]$, $e\equiv^{H}_{\lambda}e^{\prime}$, and
$e\not\equiv^{H}_{\lambda-1}e^{\prime}$, then $e\cap e^{\prime}\subseteq
V_{A_{\lambda}}(H)$.

In view of (0 ) it only remains to be shown that all edges $e,e^{\prime}\in
E(H)$ with $e\equiv^{H}_{1}e^{\prime}$ and $e\not\equiv^{H}_{0}e^{\prime}$
satisfy $e\cap e^{\prime}\subseteq V_{A_{1}}(H)$. Due to
$e\sim^{H}_{m}e^{\prime}$ there exists a smallest integer $\mu\in[0,m]$ such
that $e\sim^{H}_{\mu}e^{\prime}$. Since $e\not\sim^{H}_{0}e^{\prime}$, we have
$\mu>0$, and thus the minimality of $\mu$ yields
$e\not\sim^{H}_{\mu-1}e^{\prime}$. So (0 ) reveals $e\cap e^{\prime}\subseteq
V_{B_{\mu}}(H)$ and because of $B_{\mu}\subseteq A_{1}$ the desired inclusion
$e\cap e^{\prime}\subseteq V_{A_{1}}(H)$ follows.

This concludes our discussion of parameters and we proceed with the three
displayed properties of the train system
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$. As the train construction $\Psi^{m}$
exemplifies the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$,
the partition relation
$\mathscr{H}\longrightarrow(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})_{r}$ is clear.

Moreover, we have

$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\bullet},\mathscr{H}_{\bullet}^{+})>(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}\,.$
(12.5)

According to Definition 10.10 this implies

  1. $\bullet$

$\mathfrak{Girth}(H,\equiv^{H}_{0},\mathscr{H}^{+})>g$,

  2. $\bullet$

$\mathfrak{Girth}(H,\equiv^{H}_{\lambda},\mathscr{H}^{+})>g_{\lambda}$ for
every $\lambda\in[\ell]$,

  3. $\bullet$

and $\mathfrak{Girth}(H,\equiv^{H}_{\ell+1},\mathscr{H}^{+})>1$,

which in turn yields
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})>(g)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
By Corollary 10.11 and (12.5) we have
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\bullet})>(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
So $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\bullet},\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle B\hfil$\crcr}}})$
is an $m$-revision of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
and $\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}})>(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
follows from Fact 12.11. ∎

This concludes the first step of our proof of Proposition 10.15, i.e., the
conversion of the given  $\diamondsuit$ -principles into partite lemmata
$\Psi^{m}$ applicable to certain trains
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ of
height $\ell+1$.

###  12.6. Train constituents

Recall that our initial motivation for introducing trains was the observation
that partite constructions produce trains; moreover trains seem to offer a
chance to handle hypergraphs of the next larger girth in the context of an
argument by induction. In some sense, this subsection is the place of the
whole article where all these hopes materialise.

######  Lemma 12.13.

There exists a $((g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-amenable partite lemma $\Xi$.

This partite lemma $\Xi$ will be obtained by a diagonal variation of the
partite construction method. Vertically we use an arbitrary Ramsey
construction $\Phi$ for
$((g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-trains (see Corollary 12.8). Horizontally the basic
observation is that it was never written into stone that one has to employ the
very same partite lemma in each stage of the iterative procedure. Rather, it
is reasonable to adjust to the increasing complexity of the arising pictures
by using more and more sophisticated partite lemmata as the construction
progresses. In fact, we plan to use $\Psi^{m}$ when we need a partite lemma
for the $m^{\mathrm{th}}$ time, so a suggestive notation for the construction
we are about to describe could be
$\Xi=\mathrm{PC}\bigl{(}\Phi,(\Psi^{m})_{m\in{\mathds{N}}}\bigr{)}$. Here is
the picturesque statement we shall iterate.

######  Lemma 12.14.

Let $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}},\mathscr{G})$ be a $k$-partite $k$-uniform train system of
height $\ell+1$ with parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
whose underlying hypergraph $G$ is linear. Suppose further that

$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}},\mathscr{Q},\psi_{\Sigma})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}},\mathscr{P},\psi_{\Pi})\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr+\cr\times\crcr}}}}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$

holds for two train pictures $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}},\mathscr{P},\psi_{\Pi})$,
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}},\mathscr{Q},\psi_{\Sigma})$
of height $\ell+1$ with parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
and a $k$-partite $k$-uniform train system
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ of height $\ell+1$ such that

$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}},\mathscr{Q}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\quad\text{ and }\quad\mathrm{Girth}(H,\mathscr{H}^{+})>g\,.$

If this amalgamation occurs over the edge $e\in E(G)$,

  1. $\bullet$

another edge $e_{\star}\in E(G)$ is distinct to $e$,

  2. $\bullet$

and the constituent $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}$
is $\alpha$-revisable for some $\alpha\in{\mathds{N}}$,

then $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}$
is $(\alpha+1)$-revisable.

###### Proof.

Write $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
A\hfil$\crcr}}}=(A_{1},\dots,A_{\ell+1})$ and recall that according to Fact
12.2 the parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
D\hfil$\crcr}}}=(D_{1},\dots,D_{\ell+1})$ of the constituent
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}$
is given by $D_{\lambda}=e_{\star}\cap V_{A_{\lambda}}(G)$ for every
$\lambda\in[\ell+1]$. Let $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}_{\bullet},\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle B\hfil$\crcr}}})$
be an $\alpha$-revision of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}=(\Pi^{e_{\star}},\equiv^{\Pi}_{0},\dots,\equiv^{\Pi}_{\ell+1})$,
where

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}_{\bullet}=(\Pi^{e_{\star}},\equiv^{\Pi}_{0},\sim^{\Pi}_{1},\dots,\sim^{\Pi}_{\alpha-1},\equiv^{\Pi}_{1},\dots,\equiv^{\Pi}_{\ell+1})\quad\text{
and }\quad\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}=(B_{1},\dots,B_{\alpha})\in\powerset(D_{1})^{\alpha}\,.$

Our task is to exhibit an $(\alpha+1)$-revision of the new constituent

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}=(\Sigma^{e_{\star}},\equiv^{\Sigma}_{0},\dots,\equiv^{\Sigma}_{\ell+1})\,.$

For every standard copy $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\circ}$
of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}$
in $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}$
we can copy the above revision of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}$
onto the constituent $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}_{\circ}$,
thus getting a train

$(\Pi^{e_{\star}}_{\circ},\equiv^{\Pi_{\circ}}_{0},\sim^{\Pi_{\circ}}_{1},\dots,\sim^{\Pi_{\circ}}_{\alpha-1},\equiv^{\Pi_{\circ}}_{1},\dots,\equiv^{\Pi_{\circ}}_{\ell+1})\,.$
(12.6)

It will be convenient to set $\sim^{\Pi_{\circ}}_{0}=\equiv^{\Pi_{\circ}}_{0}$
and $\sim^{\Pi_{\circ}}_{\alpha}=\equiv^{\Pi_{\circ}}_{1}$ for every such
standard copy. The edge set of the new constituent $\Sigma^{e_{\star}}$ is the
disjoint union of the edge sets of all $\Pi_{\circ}^{e_{\star}}$ as
$\Pi_{\circ}$ varies over the standard copies of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}$
in $\Sigma$. We can thus define a quasitrain

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}_{\bullet}=(\Sigma^{e_{\star}},\equiv^{\Sigma}_{0},\sim^{\Sigma}_{1},\dots,\sim^{\Sigma}_{\alpha},\equiv^{\Sigma}_{1},\dots,\equiv^{\Sigma}_{\ell+1})$

of height $\ell+\alpha+1$ by declaring for every $\mu\in[\alpha]$ and any two
edges $e^{\prime},e^{\prime\prime}\in E(\Sigma^{e_{\star}})$ that the
statement $e^{\prime}\sim^{\Sigma}_{\mu}e^{\prime\prime}$ means: there is a
common standard copy $\Pi_{\circ}$ containing $e^{\prime}$,
$e^{\prime\prime}$, and $e^{\prime}\sim_{\mu}^{\Pi_{\circ}}e^{\prime\prime}$
holds. Next we define a set $B_{\alpha+1}$ by setting

$B_{\alpha+1}=\begin{cases}e\cap e_{\star}&\text{ if
$e\equiv^{G}_{1}e_{\star}$}\cr\varnothing&\text{ if
$e\not\equiv^{G}_{1}e_{\star}$.}\end{cases}$

We shall eventually show that
$\bigl{(}\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}_{\bullet},\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}\circ(B_{\alpha+1})\bigr{)}$ is an $(\alpha+1)$-revision of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}$.

######  Claim 12.15.

We have $B_{\alpha+1}\subseteq D_{1}$ and $|B_{\alpha+1}|\leq 1$.

###### Proof.

If $B_{\alpha+1}=\varnothing$ both assertions are clear, so we may assume
$B_{\alpha+1}=e\cap e_{\star}$ and $e\equiv^{G}_{1}e_{\star}$ from now on. The
linearity of $G$ implies $|B_{\alpha+1}|\leq 1$. We also know $e\cap
e_{\star}\subseteq V_{A_{1}}(G)$, because
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle G\hfil$\crcr}}}$
has the parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$.
Thus we have indeed

$B_{\alpha+1}=e\cap e_{\star}\subseteq e_{\star}\cap
V_{A_{1}}(G)=D_{1}\,.\qed$

######  Claim 12.16.

The sequence $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}\circ(B_{\alpha+1},D_{2},\dots,D_{\ell+1})$ parametrises
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}_{\bullet}$.

###### Proof.

For every standard copy $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\circ}$
of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}$
in $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}$
the train (12.6) has the same parameter as
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}_{\bullet}$,
i.e., $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}\circ(D_{2},\dots,D_{\ell+1})$ and, therefore, the statements
involving the sets $B_{1},\dots,B_{\alpha}$ hold. Similarly, the claims on
$D_{2},\dots,D_{\ell+1}$ follow from
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
being the parameter of the entire train
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}$
and Fact 12.2.

So it remains to be shown that if two edges $e^{\prime},e^{\prime\prime}\in
E(\Sigma^{e_{\star}})$ satisfy $e^{\prime}\equiv^{\Sigma}_{1}e^{\prime\prime}$
but $e^{\prime}\not\sim_{\alpha}^{\Sigma}e^{\prime\prime}$, then
$e^{\prime}\cap e^{\prime\prime}\in V_{B_{\alpha+1}}(\Sigma^{e_{\star}})$.
Notice that this can only happen if $e^{\prime}$ and $e^{\prime\prime}$ are in
distinct standard copies of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}$.
So in view of Lemma 7.3( ) ‣ 7.3( )( ) ‣ ( ) ‣ 7.3 there needs to exist, in
particular, an edge $e_{0}\in E(H)$ satisfying
$e^{\prime}\equiv^{\Sigma}_{1}e_{0}\equiv^{\Sigma}_{1}e^{\prime\prime}$. Now
Definition 12.1( ) ‣ 12.1 discloses
$e=\psi_{\Sigma}(e_{0})\equiv^{G}_{1}\psi_{\Sigma}(e^{\prime})=e_{\star}$,
which in turn leads to $B_{\alpha+1}=e\cap e_{\star}$. Since $\psi_{\Sigma}$
projects all vertices of $e^{\prime}\cap e^{\prime\prime}$, if there exist
any, into $e\cap e_{\star}$, this proves that $e^{\prime}\cap
e^{\prime\prime}\in V_{B_{\alpha+1}}(\Sigma^{e_{\star}})$ is indeed true. ∎

Now it remains to be shown that
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}_{\bullet})>(g)^{\alpha+1}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
Most parts of this claim are straightforward consequences of
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}_{\bullet})>(g)^{\alpha}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and of $\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}},\mathscr{Q}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.
In fact, the only part requiring some attention is that if $W$ denotes an
$(\alpha+1)$-wagon of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Sigma\hfil$\crcr}}}^{e_{\star}}_{\bullet}$,
i.e., a wagon of $(\Sigma^{e_{\star}},\equiv^{\Sigma}_{1})$, then
$\mathfrak{girth}(W,\sim_{\alpha}^{W})>g$.

Assume contrariwise that for some $n\in[2,g]$ there is an $n$-cycle

$\mathscr{C}=W_{1}v_{1}\ldots W_{n}v_{n}\,,$

where $W_{1},\dots,W_{n}$ denote wagons of $(W,\sim_{\alpha}^{W})$ and
$v_{1},\dots,v_{n}$ are vertices. Due to the definition of
$\sim^{\Sigma}_{\alpha}$ there exists for every
$t\in{\mathds{Z}}/n{\mathds{Z}}$ a standard copy
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{t}$
of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}$
such that the wagon $W_{t}$ is entirely contained in
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}^{e_{\star}}_{t}$.
Moreover, $W_{t}\neq W_{t+1}$ yields
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{t}\neq\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{t+1}$
and due to $v_{t}\in V(W_{t})\cap V(W_{t+1})$ we have $v_{t}\in V(H)$. We thus
arrive at a cycle of copies

$\mathscr{D}=\Pi^{e}_{1}v_{1}\ldots\Pi^{e}_{n}v_{n}$

in $(H,\mathscr{H}^{+})$ whose length is $n$ and all of whose connectors are
vertices sitting on the same music line of $\Sigma$ (namely the line projected
to the unique vertex in $e\cap e_{\star}$). In particular, $\mathscr{D}$ is
tidy and none of its copies is collapsible, meaning that $\mathscr{D}$ has no
master copy. But because of $\mathrm{ord}(\mathscr{D})=n\leq g$ this
contradicts $\mathrm{Girth}(H,\mathscr{H}^{+})>g$. ∎

###### Proof of Lemma 12.13.

Suppose that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
a $k$-partite $k$-uniform train of height $\ell+1$ with

$\mathfrak{girth}(F)>(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
(12.7)

and parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$,
and that $r$ is a number of colours. Pick a Ramsey construction $\Phi$ for
$((g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-trains (cf. Corollary 12.8) and set
$\Phi_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}},\mathscr{G})$. Recall that
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle G\hfil$\crcr}}}$ is
a linear train of height $\ell+1$ with parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
and that the copies in $\mathscr{G}$ are strongly induced. Without loss of
generality we can assume that all edges of $G$ belong to some copy in
$\mathscr{G}$.

Let $E(G)=\\{e(1),\ldots,e(N)\\}$ enumerate the edges of $G$. We intend to
construct recursively a sequence of train pictures
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha},\mathscr{P}_{\alpha},\psi_{\alpha})_{0\leq\alpha\leq
N}$ over $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}},\mathscr{G})$ that starts with picture zero. This is to be
done in such a way that for every $\alpha\in[0,N]$ the $\alpha^{\mathrm{th}}$
picture has the properties

  1. ( )

$\mathrm{Girth}(\Pi_{\alpha},\mathscr{P}_{\alpha}^{+})>(g+1,g+1)$;

  2. ( )

$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha},\mathscr{P}_{\alpha}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$;

  3. ( )

and for every $\beta\in(\alpha,N]$ the constituent
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha}^{e(\beta)}$
is $(\alpha+1)$-revisable.

Before we can move any further we need to check that picture zero has these
properties for $\alpha=0$. Fact 11.9 and (12.7) imply
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{0},\mathscr{P}_{0}^{+})>(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$,
which entails ( ) ‣ 12.6 immediately and in view of Lemma 8.18 clause ( ) ‣
12.6 follows as well. Furthermore, Fact 12.10 yields ( ) ‣ 12.6.

Now let any positive $\alpha\leq N$ be given for which we have already managed
to reach a train picture $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1},\mathscr{P}_{\alpha-1},\psi_{\alpha-1})$
satisfying ( ) ‣ 12.6, ( ) ‣ 12.6, and ( ) ‣ 12.6 for $\alpha-1$ instead of
$\alpha$. As a consequence of the last condition we know that its constituent
$\Pi_{\alpha-1}^{e(\alpha)}$ is $\alpha$-revisable. So Lemma 12.12 provides a
train system

$\Psi^{\alpha}_{r}\bigl{(}\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1}^{e(\alpha)}\bigr{)}=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha},\mathscr{H}_{\alpha})$

with the correct parameter satisfying

$\mathscr{H}_{\alpha}\longrightarrow\bigl{(}\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1}^{e(\alpha)}\bigr{)}_{r}\,,\quad\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha})>(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}},\quad\text{ and
}\quad\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha},\mathscr{H}_{\alpha}^{+})>(g)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}\,.$

In view of Lemma 12.3 the quasitrain picture

$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha},\mathscr{P}_{\alpha},\psi_{\alpha})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha-1},\mathscr{P}_{\alpha-1},\psi_{\alpha-1})\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr+\cr\times\crcr}}}}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha},\mathscr{H}_{\alpha})$

is actually a train picture over
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}},\mathscr{G})$. We need to check that it has the properties ( )
‣ 12.6 – ( ) ‣ 12.6.

Observe first that
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha})>(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$,
our assumption $\inf(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})>g$, and Lemma 10.8 imply

$\mathrm{girth}(H_{\alpha})>g+1\,.$ (12.8)

Moreover, $\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}_{\alpha},\mathscr{H}_{\alpha}^{+})>(g)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
contains the information
$\mathfrak{Girth}(H_{\alpha},\equiv^{H_{\alpha}}_{0},\mathscr{H}_{\alpha}^{+})>g$,
which in view of Lemma 8.18 leads to
$\mathrm{Girth}(H_{\alpha},\mathscr{H}_{\alpha}^{+})>g$. Together with our
induction hypothesis
$\mathrm{Girth}(\Pi_{\alpha-1},\mathscr{P}_{\alpha-1}^{+})>(g+1,g+1)$ and
(12.8) this allows us to apply Lemma 5.8 with $g+1$ here in place of $g$
there. This proves ( ) ‣ 12.6.

Clause ( ) ‣ 12.6 follows from Lemma 12.4. For the verification of ( ) ‣ 12.6
we consider an arbitrary integer $\beta\in(\alpha,N]$. We already know that
$\Pi_{\alpha-1}^{e(\beta)}$ is $\alpha$-revisable and we want to deduce the
$(\alpha+1)$-revisability of $\Pi_{\alpha}^{e(\beta)}$ with the help of Lemma
12.14. Its assumptions
${\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{\alpha},\mathscr{P}_{\alpha}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
and $\mathrm{Girth}(H_{\alpha},\mathscr{H}_{\alpha}^{+})>g$ have already been
established and $e(\beta)\neq e(\alpha)$ is clear. Thereby ( ) ‣ 12.6 has been
confirmed as well and the partite construction continues.

Eventually we arrive at a final picture
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{N},\mathscr{P}_{N},\psi_{N})$,
which due to ( ) ‣ 12.6 and ( ) ‣ 12.6 satisfies

$\mathrm{Girth}(\Pi_{N},\mathscr{P}_{N}^{+})>(g+1,g+1)\quad\text{ as well as
}\quad\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{N},\mathscr{P}_{N}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}\,.$

Thus the partite lemma $\Xi$ defined by
$\Xi_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle\Pi\hfil$\crcr}}}_{N},\mathscr{P}_{N}^{+})$
is $((g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-amenable. ∎

It is now immediate from Lemma 12.6 that the construction
$\mathrm{PC}(\Phi,\Xi)$, where $\Phi$ again denotes a Ramsey construction for
$((g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})$-trains, exemplifies
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.
This concludes the proof of Proposition 10.15. In the light of our earlier
work Corollary 10.18, Theorem 10.19, and Theorem 1.4 have thereby been
demonstrated as well (see Summary 10.20).

##  13\. Paradise

This section is concerned with the proofs of Theorem 1.7 and Theorem 1.6. To
this end we need a connection between $\mathrm{Girth}$ and forests of copies,
which is established in §13.1. The main result there (Lemma 13.11) asserts
that large $\mathrm{Girth}$ forces small sets of copies to be forests. In
§13.2 this lemma will allow us to analyse a partite construction utilising the
construction $\Omega^{(g)}$ provided by Theorem 10.19 as a partite lemma. As a
result, we shall prove Theorem 1.7 in a very strong form (Theorem 13.12). The
deduction of Theorem 1.6 will then be straightforward.

###  13.1. Forests of copies

Let us recall that forests of copies were introduced in Definition 1.5 as
being certain systems $\mathscr{N}\subseteq\binom{H}{F}$ consisting of
mutually isomorphic copies.

Marginally generalising the underlying setup, we shall henceforth say for a
linear system of hypergraphs $(H,\mathscr{N})$ that $\mathscr{N}$ is a forest
of copies if one can write $\mathscr{N}=\\{F_{1},\ldots,F_{|\mathscr{N}|}\\}$
in such a way that for every $j\in[2,|\mathscr{N}|]$ the set
$z_{j}=V(F_{j})\cap\bigl{(}\bigcup_{i<j}V(F_{i})\bigr{)}$ is either an edge in
$E(F_{j})\cap\bigl{(}\bigcup_{i<j}E(F_{i})\bigr{)}$ or it consists of at most
one vertex.  In this situation, $(F_{1},\ldots,F_{|\mathscr{N}|})$ is called
an admissible enumeration of $\mathscr{N}$.  It may be helpful to point out
that forests of copies are thereby allowed to contain both edge copies and
real copies.

Most results in this subsection can be regarded as appropriate adaptations of
well-known facts on ordinary forests—a notion briefly recapitulated
immediately before Theorem 1.1. To distinguish such forests from forests of
copies we shall call them edge forests in the discussion that follows.  Here
is a list of four standard facts on edge forests.

  1. ( )

Every subset of an edge forest is again an edge forest.

  2. ( )

The result of gluing two edge forests together along a vertex or an edge is
again an edge forest.

  3. ( )

If a hypergraph $H=(V,E)$ satisfies $\mathrm{girth}(H)>|E|$, then $E$ is an
edge forest.

  4. ( )

Every edge forest consisting of at least two edges has at least two leaves
(belonging to different edges).

Now we already saw in the introduction that forests of copies are not closed
under taking subsets, meaning that the naïve generalisation of ( ) ‣ 13.1
fails (see Figure 1.1). Nevertheless, we shall show in Lemma 13.4 that
deleting edge copies preserves being a forest of copies. In Lemma 13.5 and
Lemma 13.6 we shall then see that ( ) ‣ 13.1 generalises in the obvious way.
Statement ( ) ‣ 13.1 remains valid if we replace lower case $\mathrm{girth}$
by capital $\mathrm{Girth}$ (see Lemma 13.11). Finally, when dealing with ( )
‣ 13.1 we shall work with the following analogue of edges acting as “leaves”.

######  Definition 13.1.

Given a forest of copies $\mathscr{N}$ and a copy $F_{\star}\in\mathscr{N}$ we
say that $F_{\star}$ is terminal in $\mathscr{N}$ if there exists an
admissible enumeration $(F_{1},\ldots,F_{|\mathscr{N}|})$ of $\mathscr{N}$
such that $F_{\star}=F_{|\mathscr{N}|}$.

For instance, if $\mathscr{N}$ denotes a forest consisting of two copies, then
both enumerations of $\mathscr{N}$ are admissible and, consequently, both
copies in $\mathscr{N}$ are terminal. Now the variant of ( ) ‣ 13.1 we have
been alluding to reads as follows.

######  Lemma 13.2.

Every forest of at least two copies possesses at least two terminal copies.

In the argument that follows and at several other occasions occurring in the
rest of this section it will be convenient to employ the following notation.
Given a set of hypergraphs $\mathscr{N}$, we shall write $V(\mathscr{N})$ for
$\bigcup_{F_{\star}\in\mathscr{N}}V(F_{\star})$ and, similarly,
$E(\mathscr{N})$ will abbreviate
$\bigcup_{F_{\star}\in\mathscr{N}}E(F_{\star})$.

###### Proof of Lemma 13.2.

We argue by induction on the size $|\mathscr{N}|$ of the forest of copies
under consideration. The base case $|\mathscr{N}|=2$ has already been studied.
For the induction step we suppose that $\mathscr{N}$ is a forest consisting of
$n\geq 3$ copies and that every forest of copies $\mathscr{N}^{\prime}$ with
$|\mathscr{N}^{\prime}|=n-1$ has at least two terminal copies.

Since $\mathscr{N}$ admits an admissible enumeration, it possesses some
terminal copy $F_{\star}$. Notice that the set

$z_{\star}=V(F_{\star})\cap V(\mathscr{N}\smallsetminus\\{F_{\star}\\})$

is either an edge in $E(F_{\star})\cap
E(\mathscr{N}\smallsetminus\\{F_{\star}\\})$ or it consists of at most one
vertex. Therefore there exists a copy
$F_{\circ}\in\mathscr{N}\smallsetminus\\{F_{\star}\\}$ such that

  1. $\bullet$

$z_{\star}\subseteq V(F_{\circ})$

  2. $\bullet$

and if $z_{\star}$ is an edge, then $z_{\star}\in E(F_{\circ})$.

The terminality of $F_{\star}$ implies that
$\mathscr{N}\smallsetminus\\{F_{\star}\\}$ is a forest of copies and the
induction hypothesis yields an admissible enumeration $(F_{1},\ldots,F_{n-1})$
of $\mathscr{N}\smallsetminus\\{F_{\star}\\}$ whose terminal copy is distinct
from $F_{\circ}$. We contend that $F_{n-1}$ is terminal in $\mathscr{N}$ as
well and that the enumeration

$(F_{1},\ldots,F_{n-2},F_{\star},F_{n-1})$ (13.1)

exemplifies this fact. The claims that we need to check concerning the copies
$F_{2},\ldots,F_{n-2}$ follow immediately from the admissibility of
$(F_{1},\ldots,F_{n-1})$, and the claim on $F_{\star}$ is a consequence of
$F_{\circ}\in\\{F_{1},\ldots,F_{n-2}\\}$. It remains to verify that the set

$z^{+}=V(F_{n-1})\cap V\bigl{(}\\{F_{1},\ldots,F_{n-2},F_{\star}\\}\bigr{)}$

is either an edge in $E(F_{n-1})\cap
E\bigl{(}\\{F_{1},\ldots,F_{n-2},F_{\star}\\}\bigr{)}$ or that it consists of
at most one vertex. Due to the admissibility of $(F_{1},\ldots,F_{n-1})$ we
know that the set

$z^{-}=V(F_{n-1})\cap V\bigl{(}\\{F_{1},\ldots,F_{n-2}\\}\bigr{)}$

has analogous properties and thus it suffices to prove $z^{+}=z^{-}$. Because
of

$V(F_{n-1})\cap V(F_{\star})\subseteq V(F_{n-1})\cap z_{\star}\subseteq
V(F_{n-1})\cap V(F_{\circ})\subseteq z^{-}$

and

$z^{+}=z^{-}\cup\bigl{(}V(F_{n-1})\cap V(F_{\star})\bigr{)}$

this is indeed the case. Altogether we have thereby proved that the
enumeration (13.1) is admissible. Hence $F_{\star}$ and $F_{n-1}$ are two
terminal copies of $\mathscr{N}$, and the induction step is complete. ∎

The concept “dual” to a terminal copy is that of an initial copy, by which we
mean a copy in a forest that is capable of standing in the first position of
an admissible enumeration.

######  Lemma 13.3.

In a forest of copies every copy is initial.

###### Proof.

Again we argue by induction on the size of the forest under consideration. In
the base case, when the forest consists of at most two copies, every
enumeration is admissible and, therefore, every copy is initial.

For the induction step we look at a forest of copies $\mathscr{N}$ with
$n=|\mathscr{N}|\geq 3$ and at an arbitrary copy $F_{1}\in\mathscr{N}$. Lemma
13.2 discloses that $\mathscr{N}$ has a terminal copy $F_{n}\neq F_{1}$. Now
$\mathscr{N}^{\prime}=\mathscr{N}\smallsetminus\\{F_{n}\\}$ is again a forest
of copies and due to the induction hypothesis $F_{1}$ is an initial copy of
$\mathscr{N}^{\prime}$, i.e., there is an admissible enumeration
$(F_{1},\ldots,F_{n-1})$ of $\mathscr{N}^{\prime}$ that starts with $F_{1}$.
Clearly $(F_{1},\ldots,F_{n})$ is an admissible enumeration of $\mathscr{N}$
and, consequently, $F_{1}$ is indeed an initial copy of $\mathscr{N}$. ∎

Next we address “subforests” of copies obtained by the removal of edge copies.

######  Lemma 13.4.

The result of deleting an edge copy from a forest of copies is again a forest
of copies.

###### Proof.

Let $\mathscr{N}$ be a forest of copies containing some edge copy $e^{+}$. We
are to prove that $\mathscr{N}^{\prime}=\mathscr{N}\smallsetminus\\{e^{+}\\}$
is a forest of copies as well. To avoid trivialities we may suppose that
$\mathscr{N}^{\prime}\neq\varnothing$. If $e\in E(\mathscr{N}^{\prime})$ we
choose a copy $F_{1}\in\mathscr{N}^{\prime}$ satisfying $e\in E(F_{1})$ and
otherwise we let $F_{1}\in\mathscr{N}^{\prime}$ be arbitrary. By Lemma 13.3
there is an admissible enumeration

$(F_{1},\ldots,F_{i},e^{+},F_{i+1},\ldots,F_{|\mathscr{N}^{\prime}|})$

of $\mathscr{N}$ starting with $F_{1}$ and one checks immediately that the
enumeration $(F_{1},\ldots,F_{|\mathscr{N}^{\prime}|})$ of
$\mathscr{N}^{\prime}$ obtained by deleting $e^{+}$ is admissible. ∎

Roughly speaking, the lemma that follows asserts that if we glue two forests
of copies $\mathscr{A}$ and $\mathscr{B}$ together at a vertex, then the
result is again a forest of copies.

######  Lemma 13.5.

Suppose that $(H,\mathscr{N})$ is a linear system of hypergraphs admitting a
partition $\mathscr{N}=\mathscr{A}\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}}\mathscr{B}$
such that $|V(\mathscr{A})\cap V(\mathscr{B})|\leq 1$. If both $\mathscr{A}$
and $\mathscr{B}$ are forests of copies, then so is $\mathscr{N}$.

###### Proof.

The case $\mathscr{B}=\varnothing$ being clear we may suppose that
$\mathscr{B}$ contains at least one copy. Since the set $z=V(\mathscr{A})\cap
V(\mathscr{B})$ consists of at most one vertex, there exists a copy
$F^{\prime}_{1}\in\mathscr{B}$ such that $z\subseteq V(F^{\prime}_{1})$. Now
Lemma 13.3 shows that $\mathscr{B}$ has an admissible enumeration
$(F^{\prime}_{1},\ldots,F^{\prime}_{|\mathscr{B}|})$ starting with
$F^{\prime}_{1}$. It is not hard to check that if
$(F_{1},\ldots,F_{|\mathscr{A}|})$ is an arbitrary admissible enumeration of
$\mathscr{A}$, then the enumeration
$(F_{1},\ldots,F_{|\mathscr{A}|},F^{\prime}_{1},\ldots,F^{\prime}_{|\mathscr{B}|})$
of $\mathscr{N}$ is admissible as well. ∎

There is a similar statement addressing the case that the glueing occurs in an
entire edge of the underlying hypergraph.

######  Lemma 13.6.

Let $(H,\mathscr{N})$ be a linear system of hypergraphs and let
$\mathscr{N}=\mathscr{A}\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}}\mathscr{B}$
be a partition such that $V(\mathscr{A})\cap V(\mathscr{B})\subseteq e$ holds
for some edge $e$ of $H$. If both $\mathscr{A}\cup\\{e^{+}\\}$ and
$\mathscr{B}\cup\\{e^{+}\\}$ are forests of copies, then so is $\mathscr{N}$.

###### Proof.

Take an arbitrary admissible enumeration $(F_{1},\ldots,F_{a})$ of
$\mathscr{A}\cup\\{e^{+}\\}$, as well as an admissible enumeration
$(e^{+},F^{\prime}_{2},\ldots,F^{\prime}_{b})$ of
$\mathscr{B}\cup\\{e^{+}\\}$, where $a=|\mathscr{A}\cup\\{e^{+}\\}|$ and
$b=|\mathscr{B}\cup\\{e^{+}\\}|$, respectively. Now
$(F_{1},\ldots,F_{a},F^{\prime}_{2},\ldots,F^{\prime}_{b})$ is an admissible
enumeration of $\mathscr{N}\cup\\{e^{+}\\}$. If $e^{+}\in\mathscr{N}$ this
proves that $\mathscr{N}$ is indeed a forest of copies and in case
$e^{+}\not\in\mathscr{N}$ Lemma 13.4 leads us to the same conclusion. ∎

An iterative application of the next result will allow us to relate forests of
copies to $\mathrm{Girth}$.

######  Lemma 13.7.

Given a linear system of hypergraphs $(H,\mathscr{N})$ with
${\mathrm{Girth}(H,\mathscr{N}^{+})>|\mathscr{N}|\geq 2}$ there exists a copy
$F_{\star}\in\mathscr{N}$ such that the set

$z_{\star}=V(F_{\star})\cap V(\mathscr{N}\smallsetminus\\{F_{\star}\\})$
(13.2)

is either an edge in $E(F_{\star})\cap
E(\mathscr{N}\smallsetminus\\{F_{\star}\\})$ or it consists of at most one
vertex.

###### Proof.

Assume for the sake of contradiction that no such copy $F_{\star}$ exists. We
aim at building a tidy cycle of copies in $(H,\mathscr{N}^{+})$ consisting of
at most $|\mathscr{N}|$ distinct copies, which does not possess a master copy.
Looking at the potential connectors of such a cycle, we call a vertex $x\in
V(H)$ useful if there exist distinct copies
$F^{\prime},F^{\prime\prime}\in\mathscr{N}$ such that $x\in V(F^{\prime})\cap
V(F^{\prime\prime})$. Similarly, an edge $e\in E(H)$ is said to be useful if
there are distinct copies $F^{\prime},F^{\prime\prime}\in\mathscr{N}$
satisfying $e\in E(F^{\prime})\cap E(F^{\prime\prime})$. For instance, for
every copy $F_{\star}\in\mathscr{N}$ all vertices belonging to the
corresponding set $z_{\star}$ defined in (13.2) are useful.

######  Claim 13.8.

For every useful vertex $x$ there are two distinct copies
$F^{\prime}_{x},F^{\prime\prime}_{x}\in\mathscr{N}$ such that

  1. ( )

$x\in V(F^{\prime}_{x})\cap V(F^{\prime\prime}_{x})$;

  2. ( )

and each of $F^{\prime}_{x}$, $F^{\prime\prime}_{x}$ has at most one useful
edge containing $x$.

###### Proof.

Given $x$ we consider an auxiliary set system $S_{x}$ with vertex set

$V(S_{x})=\bigl{\\{}F_{\star}\in\mathscr{N}\colon x\in
V(F_{\star})\bigr{\\}}\,.$

For every useful edge $e$ containing $x$ the set
$\varphi_{e}=\bigl{\\{}F_{\star}\in V(S_{x})\colon e\in
E(F_{\star})\bigr{\\}}$ has at least the size $2$. Thus we can define the edge
set of our set system $S_{x}$ by

$E(S_{x})=\bigl{\\{}\varphi_{e}\colon\text{$e$ is useful and $x\in
e$}\bigr{\\}}\,.$

We are to prove that $S_{x}$ has at least two vertices whose degree is at most
one. Due to $v(S_{x})\geq 2$ the failure of this statement would imply that
$S_{x}$ contains some cycle $\varphi_{e(1)}F_{1}\ldots\varphi_{e(n)}F_{n}$ (in
the sense of Definition 1.3). But then $F_{1}e(2)\ldots F_{n}e(1)$ is a tidy
cycle of copies in $(H,\mathscr{N})$ all of whose connectors are edges, which
in view of

$\mathrm{Girth}(H,\mathscr{N}^{+})>|\mathscr{N}|\geq v(S_{x})\geq n$

contradicts Lemma 4.15. ∎

Next we shall construct a cycle of copies in $(H,\mathscr{N})$ which has only
vertex connectors and some further special property. We commence by examining
desirable subsequences of the form $q^{\prime}F_{\star}q^{\prime\prime}$ of
the envisaged cycle. Given a copy $F_{\star}\in\mathscr{N}$ and distinct
useful vertices $q^{\prime},q^{\prime\prime}\in V(F_{\star})$ we say that
$F_{\star}$ is secure between $q^{\prime}$ and $q^{\prime\prime}$ provided the
following statement holds: If there is an edge $f$ satisfying
$q^{\prime},q^{\prime\prime}\in f\in E(F_{\star})$, then this edge is not
useful. The intuition here is that it is impossible to collapse copies that
occur securely between their neighbouring connectors.

######  Claim 13.9.

If $F_{\star}\in\mathscr{N}$ and $q^{\prime}\in V(F_{\star})$ is useful, then
there exists a copy
$F_{\star\star}\in\mathscr{N}\smallsetminus\\{F_{\star}\\}$ satisfying
$q^{\prime}\in V(F_{\star\star})$ together with another useful vertex
$q^{\prime\prime}\in V(F_{\star\star})$ such that $F_{\star\star}$ is secure
between $q^{\prime}$ and $q^{\prime\prime}$.

###### Proof.

Let $F^{\prime}_{q^{\prime}},F^{\prime\prime}_{q^{\prime}}\in\mathscr{N}$ be
the copies Claim 13.8 provides for $x=q^{\prime}$, choose
$F_{\star\star}\in\\{F^{\prime}_{q^{\prime}},F^{\prime\prime}_{q^{\prime}}\\}$
such that $F_{\star\star}\neq F_{\star}$, and observe that Claim 13.8( ) ‣
13.8 ensures $q^{\prime}\in V(F_{\star\star})$. Let the set $z^{\star\star}$
be defined with respect to $F_{\star\star}$ as in (13.2). The failure of our
lemma entails $|z^{\star\star}|\geq 2$ and thus there exists a vertex
$q^{\circ}\in z^{\star\star}$ distinct from $q^{\prime}$. If $F_{\star\star}$
is secure between $q^{\prime}$ and $q^{\circ}$, we can just take
$q^{\prime\prime}=q^{\circ}$, so suppose from now on that this is not the
case. This means that there exists a useful edge $f$ satisfying
$q^{\prime},q^{\circ}\in f\in E(F_{\star\star})$.

$f$$f^{\prime}$$F_{\star}$$F_{\star\star}$$q^{\circ}$$q^{\prime}$$q^{\prime\prime}$
Figure 13.1. Security of $F_{\star\star}$

Clearly $f\subseteq z^{\star\star}$ and by appealing to the failure of our
lemma again we learn $z^{\star\star}\neq f$. Hence there exists a vertex
$q^{\prime\prime}\in z^{\star\star}\smallsetminus f$ and it suffices to prove
that $F_{\star\star}$ is secure between $q^{\prime}$ and $q^{\prime\prime}$
(see Figure 13.1). If this were not the case, there had to exist a useful edge
$f^{\prime}$ such that $q^{\prime},q^{\prime\prime}\in f^{\prime}\in
E(F_{\star\star})$. Now $q^{\prime\prime}\in f^{\prime}\smallsetminus f$
yields $f\neq f^{\prime}$ and, therefore, $F_{\star\star}$ violates clause ( )
‣ 13.8 of Claim 13.8. This contradiction concludes the proof of Claim 13.9. ∎

Let us call a cycle of copies $F_{1}q_{1}\ldots F_{n}q_{n}$ in
$(H,\mathscr{N})$ or $(H,\mathscr{N}^{+})$ special if

  1. $\bullet$

the copies $F_{1},\ldots,F_{n}$ are distinct,

  2. $\bullet$

the connectors $q_{1},\ldots,q_{n}$ are vertices,

  3. $\bullet$

and with at most one exception every copy $F_{i}$ is secure between $q_{i-1}$
and $q_{i}$.

######  Claim 13.10.

There exist a special cycle of copies in $(H,\mathscr{N})$.

###### Proof.

An iterative application of Claim 13.9 allows us to construct an infinite
sequence

$F_{1}q_{1}F_{2}q_{2}\ldots$

consisting of copies $F_{1},F_{2},\ldots\in\mathscr{N}$ and useful vertices
$q_{1},q_{2},\ldots,$ such that for every $i\in{\mathds{N}}$

  1. (0 )

the vertex $q_{i}$ belongs to $V(F_{i})\cap V(F_{i+1})$,

  2. (0 )

the copies $F_{i}$, $F_{i+1}$ are distinct,

  3. (0 )

and $F_{i+1}$ is secure between $q_{i}$ and $q_{i+1}$.

Indeed, let $F_{1}\in\mathscr{N}$ be arbitrary and let $q_{1}\in V(F_{1})$ be
a useful vertex. If for some natural number $m$ we have already constructed
the initial segment $F_{1}q_{1}\ldots F_{m}q_{m}$ of our infinite sequence, we
apply Claim 13.9 to $(F_{m},q_{m})$ here in place of $(F_{\star},q^{\prime})$
there, thus obtaining a copy $F_{m+1}$ and a useful vertex $q_{m+1}$ that
allow us to continue.

Since $\mathscr{N}$ is finite, there exists some $n\in{\mathds{N}}$ such that
$F_{n+1}\in\\{F_{1},\ldots,F_{n}\\}$. If $n$ denotes the least such natural
number, then $F_{1},\ldots,F_{n}$ are distinct and, hence, there is a unique
index $i\in[n]$ such that $F_{i}=F_{n+1}$. For notational simplicity we may
suppose that $i=1$. Since (0 ) implies $n\geq 2$, the cyclic sequence

$\mathscr{C}=F_{1}q_{1}\ldots F_{n}q_{n}$

has all properties of a cycle of copies except that we do not know whether its
connectors are distinct. Moreover, (0 ) tells us that $F_{1}$ is the only copy
in $\mathscr{C}$ that might be insecure.

Thus if the connectors of $\mathscr{C}$ happen to be distinct, then
$\mathscr{C}$ is the desired special cycle of copies. If $q_{1},\ldots,q_{n}$
fail to be distinct, we take a pair $(r,s)$ of indices with $1\leq r<s\leq n$,
$q_{r}=q_{s}$, and, subject to this, such that $s-r$ is minimal. Now one
checks easily that $F_{r+1}q_{r+1}\ldots F_{s}q_{s}$ is a special cycle of
copies. ∎

Throughout the rest of the proof we consider a special cycle of copies
$\mathscr{C}=F_{1}q_{1}\ldots F_{n}q_{n}$ in the extended system
$(H,\mathscr{N}^{+})$ whose length $n$ is minimal. Claim 13.10 discloses
$n\leq|\mathscr{N}|$ and, consequently, we have
$\mathrm{ord}(\mathscr{C})<\mathrm{Girth}(H,\mathscr{N}^{+})$. In other words,

  1. ( )

either $\mathscr{C}$ fails to be tidy

  2. ( )

or $\mathscr{C}$ has a master copy.

We shall show that both alternatives lead to a contradiction. Let us deal with
case ( ) ‣ 13.1 first. Since $\mathscr{C}$ has no edge connectors, this is
only possible if ($T2$) fails for some edge $f\in E(H)$. By symmetry we can
suppose that $1\in M(f)$. Write $M(f)=\\{i(1),\ldots,i(m)\\}$ with
$1=i(1)<i(2)<\ldots i(m)\leq n$ and $m\geq 2$. We observe that the cyclic
sequences

$\displaystyle\mathscr{D}_{\mu}$
$\displaystyle=f^{+}q_{i(\mu)}F_{i(\mu)+1}q_{i(\mu)+1}\ldots
F_{i(\mu+1)}q_{i(\mu+1)}\quad\text{ for }\mu\in[m-1]$ $\displaystyle\text{ and
}\quad\mathscr{D}_{m}$ $\displaystyle=f^{+}q_{i(m)}F_{i(m)+1}q_{i(m)+1}\ldots
F_{1}q_{1}$

are shorter than $\mathscr{C}$. Indeed, this is clear for $m\geq 3$ and in
case $m=2$ it follows from the fact that the two members of $M(f)$ cannot be
consecutive in ${\mathds{Z}}/n{\mathds{Z}}$.

We aim at showing that at least one of
$\mathscr{D}_{1},\ldots,\mathscr{D}_{m}$ contradicts the minimality of $n$.
Notice that the newly inserted edge copy $f^{+}$ can cause trouble in two
different ways. First, it might be insecure in all of
$\mathscr{D}_{1},\ldots,\mathscr{D}_{m}$ and, second, it might happen that one
of $\mathscr{D}_{1},\ldots,\mathscr{D}_{m}$ contains $f^{+}$ twice. However,
since $\mathscr{C}$ itself is special, at most one of
$\mathscr{D}_{1},\ldots,\mathscr{D}_{m}$ contains an insecure appearance of
one of $F_{1},\ldots,F_{n}$. Thus at most two among
$\mathscr{D}_{1},\ldots,\mathscr{D}_{m}$ can fail to contradict the minimality
of $n$, one due to containing two insecure copies and the (potential) other
one due to containing $f^{+}$ twice.

In other words, the only case where we are not done yet occurs if $m=|M(f)|=2$
and $f^{+}$ is among $F_{1},\ldots,F_{n}$. As $f^{+}=F_{i}$ implies $i-1,i\in
M(f)$, this case requires $M(f)=\\{i-1,i\\}$, which contradicts the choice of
$f$. Altogether, we have thereby shown that ( ) ‣ 13.1 is indeed impossible,
i.e., that our minimal special cycle of copies $\mathscr{C}$ in
$(H,\mathscr{N}^{+})$ is tidy.

Therefore ( ) ‣ 13.1 holds, i.e., $\mathscr{C}$ has a master copy. By symmetry
we may suppose that $F_{1}$ is a master copy of $\mathscr{C}$ and that the
family of edges $\\{f_{i}\in E(F_{1})\colon i\in[2,n]\\}$ exemplifies this
fact. For every $i\in[2,n]$ Fact 4.11 tells us $f_{i}\in E(F_{i})\cap
E(F_{1})$, wherefore the edge $f_{i}$ is useful. Combined with
$q_{i-1},q_{i}\in f_{i}$ this implies that $F_{i}$ fails to be secure between
$q_{i-1}$ and $q_{i}$. Thus despite being special $\mathscr{C}$ contains at
least $n-1$ insecure copies, which is only possible if $n=2$. But now the
useful edge $f_{2}$ exemplifies that $F_{1}$ is insecure as well and we have
obtained the final contradiction that rules out option ( ) ‣ 13.1 and thereby
concludes the proof of Lemma 13.7. ∎

######  Lemma 13.11.

Let $(H,\mathscr{N})$ be a linear system of hypergraphs. If
$\mathrm{Girth}(H,\mathscr{N}^{+})>|\mathscr{N}|$, then $\mathscr{N}$ is a
forest of copies.

###### Proof.

We argue by induction on $n=|\mathscr{N}|$, the base case $n=1$ being clear.
In the induction step we have to deal with a linear system $(H,\mathscr{N})$
consisting of $n\geq 2$ copies that satisfies
$\mathrm{Girth}(H,\mathscr{N}^{+})>|\mathscr{N}|$. Let
$F_{\star}\in\mathscr{N}$ be a copy obtained by applying Lemma 13.7 to
$\mathscr{N}$ and set
$\mathscr{N}_{\star}=\mathscr{N}\smallsetminus\\{F_{\star}\\}$. Since
$\mathrm{Girth}(H,\mathscr{N}_{\star}^{+})>|\mathscr{N}|>|\mathscr{N}_{\star}|$,
the induction hypothesis shows that $\mathscr{N}_{\star}$ is a forest of
copies. If $(F_{1},\ldots,F_{n-1})$ is an admissible enumeration of
$\mathscr{N}_{\star}$, then $(F_{1},\ldots,F_{n-1},F_{\star})$ is the desired
admissible enumeration of $\mathscr{N}$. ∎

###  13.2. The final partite construction

We shall prove the following strong form of Theorem 1.7 alluded to in the
introduction.

######  Theorem 13.12.

Let a hypergraph $F$ and a natural number $g\geq 2$ satisfy
$\mathrm{girth}(F)>g$. If $r,n\geq 2$ are two further natural numbers, then
there exists a linear system of hypergraphs $(H,\mathscr{H})$ such that

  1. $\bullet$

$\mathscr{H}\longrightarrow(F)_{r}$

  2. $\bullet$

and for every $\mathscr{N}\subseteq\mathscr{H}^{+}$ with
$|\mathscr{N}|\in[2,n]$ there exists some $\mathscr{X}\subseteq\mathscr{H}$
for which $\mathscr{N}\cup\mathscr{X}$ is a forest of copies and
$|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$.

Before we embark on the proof of this result we would like to point out that
for $n\leq g$ it follows quickly from a pair of statements that have been
obtained earlier. Indeed, the construction $\Omega^{(g)}$ (see Theorem 10.19)
delivers a linear system of hypergraphs $(H,\mathscr{H})$ such that
$\mathscr{H}\longrightarrow(F)_{r}$ and $\mathrm{Girth}(H,\mathscr{H}^{+})>g$.
Now for every $\mathscr{N}\subseteq\mathscr{H}^{+}$ with
$|\mathscr{N}|\in[2,g]$ we have
$\mathrm{Girth}(H,\mathscr{N}^{+})>g\geq|\mathscr{N}|$ and owing to Lemma
13.11 this implies that $\mathscr{N}$ is a forest of copies. In other words,
the second bullet holds for $\mathscr{X}=\varnothing$, as desired.

The general case of Theorem 13.12 will be proved by means of a further
application of the partite construction method. This will involve pictures of
the following kind.

######  Definition 13.13.

Given integers $g,n\geq 2$ a picture $(\Pi,\mathscr{P},\psi)$ is said to be
$(g,n)$-certified if each of its constituents $\Pi^{e}$ satisfies
$\mathrm{girth}(\Pi^{e})>n$ and for every
$\mathscr{N}\subseteq\mathscr{P}^{+}$ with $|\mathscr{N}|\in[2,n]$ there
exists some $\mathscr{X}\subseteq\mathscr{P}$ such that
$\mathscr{N}\cup\mathscr{X}$ is a forest of copies and
$|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$.

Let us show first that being certified is a property we can expect any
reasonable picture zero to have.

######  Lemma 13.14.

Suppose that a hypergraph $F$ and an integer $g\geq 2$ satisfy
$\mathrm{girth}(F)>g$. If $(G,\mathscr{G})$ denotes any linear system of
hypergraphs with $\mathscr{G}\subseteq\binom{G}{F}$, then the picture zero
over this system is $(g,n)$-certified for every integer $n\geq 2$.

###### Proof.

Let $(\Pi,\mathscr{P},\psi)$ denote the picture zero under discussion. Since
all of its constituents are matchings, their girth is greater than $n$ for
every $n\geq 2$. Proceeding with the second property, we suppose that any
$\mathscr{N}\subseteq\mathscr{P}^{+}$ with $|\mathscr{N}|\geq 2$ is given. For
every copy $F_{\star}\in\mathscr{P}$ we set
$\mathscr{N}(F_{\star})=\mathscr{N}\cap\bigl{(}E^{+}(F_{\star})\cup\\{F_{\star}\\}\bigr{)}$.
Since $\Pi$ is the disjoint union of the copies in $\mathscr{P}$, this
stipulation yields a partition
$\mathscr{N}=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{F_{\star}\in\mathscr{P}}\mathscr{N}(F_{\star})$.

There are two sufficient conditions ensuring that a partition class
$\mathscr{N}(F_{\star})$ is a forest of copies. First, if $F_{\star}\in
N(F_{\star})$, then every enumeration of $\mathscr{N}(F_{\star})$ starting
with $F_{\star}$ itself is admissible. Second, if
$F_{\star}\not\in\mathscr{N}(F_{\star})$ and $|\mathscr{N}(F_{\star})|\leq g$,
then $\mathscr{N}(F_{\star})$ is a forest of edge copies due to
$\mathrm{girth}(F_{\star})>g$. For these reasons, the set

$\mathscr{X}=\bigl{\\{}F_{\star}\in\mathscr{P}\colon|\mathscr{N}(F_{\star})|\geq
g+1\bigr{\\}}$

has the property that $\mathscr{N}\cup\mathscr{X}$ is a forest of copies.

So it remains to prove $|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$. The
obvious bound $|\mathscr{X}|\leq|\mathscr{N}|/(g+1)$ rewrites as
$(g-1)|\mathscr{X}|+2(|\mathscr{X}|-1)\leq|\mathscr{N}|-2$ and provided that
$\mathscr{X}$ is nonempty the desired upper bound on $|\mathscr{X}|$ follows.
Moreover, if $\mathscr{X}$ is empty, then we just need to appeal to
$|\mathscr{N}|\geq 2$. ∎

The picturesque lemma appropriate for the present context reads as follows.

######  Lemma 13.15.

Suppose that $g,n\geq 2$ are integers and that

$(\Sigma,\mathscr{Q},\psi_{\Sigma})=(\Pi,\mathscr{P},\psi_{\Pi})\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr+\cr\times\crcr}}}}(H,\mathscr{H})$
(13.3)

holds for two pictures $(\Pi,\mathscr{P},\psi_{\Pi})$ and
$(\Sigma,\mathscr{Q},\psi_{\Sigma})$ over a linear system of hypergraphs
$(G,\mathscr{G})$ and a linear $k$-partite $k$-uniform system of hypergraphs
$(H,\mathscr{H})$. If

  1. $\bullet$

$(\Pi,\mathscr{P},\psi_{\Pi})$ is $(g,n)$-certified

  2. $\bullet$

and $\mathrm{Girth}(H,\mathscr{H}^{+})>n$,

then $(\Sigma,\mathscr{Q},\psi_{\Sigma})$ is $(g,n)$-certified as well.

###### Proof.

The demand on the girth of the constituents of
$(\Sigma,\mathscr{Q},\psi_{\Sigma})$ follows easily from the corresponding
property of the constituents of $(\Pi,\mathscr{P},\psi_{\Pi})$ combined with
$\mathrm{Girth}(H,\mathscr{H}^{+})>n$ and the linearity of the vertical
hypergraph $G$. So it remains to deal with the forest extension property.

We shall say that $(\mathscr{N},\varphi)$ is a good pair if
$\mathscr{N}\subseteq\mathscr{Q}^{+}$ and
$\varphi\colon\mathscr{N}\longrightarrow\mathscr{H}^{+}$ is a map such that
for every copy $F_{\star}\in\mathscr{N}$ one of the following two cases
occurs:

  1. $\bullet$

Either $F_{\star}\in E^{+}(H)$ is an edge copy and
$\varphi(F_{\star})=F_{\star}$

  2. $\bullet$

or $\varphi(F_{\star})=\Pi_{\star}^{e}\in\mathscr{H}$ and the standard copy
$(\Pi_{\star},\mathscr{P}_{\star})$ extending $\Pi^{e}_{\star}$ satisfies
$F_{\star}\in\mathscr{P}_{\star}^{+}$.

Notice that for every $\mathscr{N}\subseteq\mathscr{Q}^{+}$ there is a map
$\varphi\colon\mathscr{N}\longrightarrow\mathscr{H}^{+}$ such that
$(\mathscr{N},\varphi)$ is a good pair. But for clarity we would like to
remark that $\varphi$ does not need to be uniquely determined by
$\mathscr{N}$, since for $f^{+}\in\mathscr{N}\cap E^{+}(H)$ there may be
several legitimate choices for $\varphi(f^{+})$.

A good pair $(\mathscr{X},\xi)$ is said to resolve another good pair
$(\mathscr{N},\varphi)$ if

  1. ( )

$\mathscr{X}\subseteq\mathscr{Q}$,

  2. ( )

$\xi[\mathscr{X}]\subseteq\varphi[\mathscr{N}]$,

  3. ( )

and $\mathscr{N}\cup\mathscr{X}$ is a forest of copies.

We shall prove the following strengthening of our claim: Every good pair
$(\mathscr{N},\varphi)$ such that $|\mathscr{N}|\in[2,n]$ is resolved by
another good pair $(\mathscr{X},\xi)$ satisfying
$|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$.

Arguing indirectly we fix a good pair $(\mathscr{N},\varphi)$ with
$|\mathscr{N}|\in[2,n]$

  1. (0 )

that is not resolved by any good pair $(\mathscr{X},\xi)$ with
$|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$

  2. (0 )

and subject to this in such a way that $\big{|}\varphi[\mathscr{N}]\big{|}$
minimal.

Assume first that $\big{|}\varphi[\mathscr{N}]\big{|}\leq 1$. Because of
$|\mathscr{N}|\geq 2$ this is only possible if there exists some standard copy
$(\Pi_{\star},\mathscr{P}_{\star})$ such that
$\mathscr{N}\subseteq\mathscr{P}_{\star}^{+}$ and
$\varphi[\mathscr{N}]=\\{\Pi^{e}_{\star}\\}$, where $\Pi_{\star}$ extends
$\Pi^{e}_{\star}$. As the picture $(\Pi,\mathscr{P},\psi_{\Pi})$ is
$(g,n)$-certified, there exists a set
$\mathscr{X}\subseteq\mathscr{P}_{\star}\subseteq\mathscr{Q}$ such that
$\mathscr{N}\cup\mathscr{X}$ is a forest of copies and
$|\mathscr{X}|\leq\frac{|\mathscr{N}|-2}{g-1}$. Let $\xi$ be the unique map
from $\mathscr{X}$ to $\\{\Pi^{e}_{\star}\\}$. Now the good pair
$(\mathscr{X},\xi)$ resolves $(\mathscr{N},\varphi)$ and thus it contradicts
(0 ). This argument proves

$\big{|}\varphi[\mathscr{N}]\big{|}\geq 2\,.$ (13.4)

Since $|\varphi[\mathscr{N}]|\leq|\mathscr{N}|\leq
n<\mathrm{Girth}(H,\mathscr{H}^{+})$, Lemma 13.11 tells us that
$\varphi[\mathscr{N}]$ is a forest of copies. We choose a terminal copy
$\Pi^{e}_{\star}$ in this forest, partition $\mathscr{N}$ into the sets

$\mathscr{A}=\varphi^{-1}(\Pi^{e}_{\star})\quad\text{ and
}\quad\mathscr{B}=\mathscr{N}\smallsetminus\mathscr{A}\,,$

and remark that (13.4) implies

$1\leq\big{|}\varphi[\mathscr{A}]\big{|},\big{|}\varphi[\mathscr{B}]\big{|}<\big{|}\varphi[\mathscr{N}]\big{|}\,.$
(13.5)

Due to $\varphi[\mathscr{A}]=\\{\Pi^{e}_{\star}\\}$ and
$\varphi[\mathscr{B}]=\varphi[\mathscr{N}]\smallsetminus\\{\Pi^{e}_{\star}\\}$
the terminal choice of $\Pi^{e}_{\star}$ guarantees that the set
$x=V(\varphi[A])\cap V(\varphi[\mathscr{B}])$ is either an edge in
$E(\varphi[A])\cap E(\varphi[\mathscr{B}])$ or it consists of at most one
vertex. We begin with the former case, as it illustrates better how the upper
bound $\frac{|\mathscr{N}|-2}{g-1}$ arises in the proof.

First Case. $x\in E(\varphi[A])\cap E(\varphi[\mathscr{B}])$.

Let us consider the sets $\mathscr{A}^{\prime}=\mathscr{A}\cup\\{x^{+}\\}$ and
$\mathscr{B}^{\prime}=\mathscr{B}\cup\\{x^{+}\\}$. We claim that there are
maps $\alpha\colon\mathscr{A}^{\prime}\longrightarrow\mathscr{H}^{+}$ and
$\beta\colon\mathscr{B}^{\prime}\longrightarrow\mathscr{H}^{+}$ such that

$\text{$(\mathscr{A}^{\prime},\alpha)$ and $(\mathscr{B}^{\prime},\beta)$ are
good pairs, $\alpha[\mathscr{A}^{\prime}]=\varphi[\mathscr{A}]$, and
$\beta[\mathscr{B}^{\prime}]=\varphi[\mathscr{B}]$}\,.$ (13.6)

Concerning the existence of $\alpha$, we observe that in case
$x^{+}\in\mathscr{A}$ we may just take the restriction of $\varphi$ to
$\mathscr{A}$. If $x^{+}\not\in\mathscr{A}$ we additionally need to specify an
appropriate value of $\alpha(x^{+})$ in $\varphi[\mathscr{A}]$, which is
possible because of $x\in E(\varphi[\mathscr{A}])$. Thus the desired map
$\alpha$ does indeed exist and we may argue similarly with respect to $\beta$.
Thereby (13.6) is proved.

Next we seek to establish

$|\mathscr{A}^{\prime}|,|\mathscr{B}^{\prime}|\in[2,n]\,.$ (13.7)

Since (13.5) implies $\mathscr{A},\mathscr{B}\neq\mathscr{N}$, the upper
bounds follow from $|\mathscr{N}|\leq n$. Assume towards a contradiction that
$|\mathscr{A}^{\prime}|\leq 1$. By $\mathscr{A}\neq\varnothing$ this is only
possible if $\mathscr{A}=\\{x^{+}\\}$, whence
$\mathscr{B}^{\prime}=\mathscr{N}$. Now $(\mathscr{N},\beta)$ is a good pair,