The Open Dihypergraph Dichotomy for ${}^\kappa\kappa$]
The open dihypergraph dichotomy for generalized
Baire spaces and its applications

Philipp Schlicht
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK

Dorottya Sziráki
Dorottya Sziráki,
Völgy u. 39c, H–1021
Budapest, Hungary

[2020](Primary) 03E15, (Secondary) 03E35, 03E55

We are grateful to Stevo Todorčević for comments and discussions.
We further thank Raphaël Carroy and Ben Miller for answering a query regarding their work.

The first-listed author was partially supported by EPSRC grant number EP/V009001/1 and FWF grant number I4039.
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 794020 of the first-listed author.
The second-listed author was partially supported by NKFIH grant number K129211.
For the purpose of open access, the authors have applied a ‘Creative Commons Attribution' (CC BY) public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

The open graph dichotomy
for a
subset $X$ of the Baire space ${}^\omega\omega$
states that any open graph on $X$ either admits a coloring in countably many colors or contains a perfect complete subgraph.
This strong version of the open graph axiom
for $X$
was introduced by Feng and Todorčević to study definable sets of reals.
We first show that its recent generalization to infinite dimensional directed hypergraphs by Carroy, Miller and Soukup
holds for all subsets of the Baire space in Solovay's model,
extending a theorem of Feng in dimension $2$.

The main theorem
lifts this result to generalized Baire spaces ${}^\kappa\kappa$ in two ways.
\beginenumerate-(1)

* For any regular infinite cardinal $\kappa$, the following
holds after a Lévy collapse of an inaccessible cardinal $\lambda>\kappa$ to $\kappa^+$.

Suppose that $H$ is a $\kappa$-dimensional box-open directed hypergraph on a subset of ${}^\kappa\kappa$ such that $H$ is definable from a $\kappa$-sequence of ordinals.
Then either $H$ admits a
coloring in $\kappa$ many colors
or there exists a continuous homomorphism from a
large directed hypergraph to $H$.

* If $\lambda$ is a Mahlo cardinal, then
the previous result extends to all  box-open directed hypergraphs on any subset of ${}^\kappa\kappa$ that is definable from a $\kappa$-sequence of ordinals.
\endenumerate-(1)

This approach is applied to solve several problems about definable subsets of generalized Baire spaces.
For instance, we obtain
variants of the Hurewicz dichotomy that characterizes subsets of $K_\sigma$ sets, strong variants of the Kechris-Louveau-Woodin dichotomy that characterizes when two disjoint sets can be separated by an $F_\sigma$ set, the determinacy of Väänänen's perfect set game for all subsets of ${}^\kappa\kappa$, an asymmetric version of the Baire property and an analogue of the Jayne-Rogers theorem that characterizes $G_\delta$-measurable functions.

§ INTRODUCTION

It is natural to wonder whether Ramsey's theorem for $n$-tuples of natural numbers <cit.> can be extended to the set of real numbers.
Sierpinski's counterexample is a partition of pairs of reals into two pieces such that no uncountable homogeneous set exists <cit.>.
Since his example is not constructive, one can ask if a version of Ramsey's theorem holds for simple partitions.

gave a partial answer to this
by proving that for any partition

of pairs of reals
into two open pieces there exists an uncountable (in fact, perfect) homogeneous set
Blass generalized this to partitions of $n$-tuples into finitely many Borel pieces

by weakening the conclusion

instead of homogeneity, one requires that
all pairs in the set belong to at most $(n-1)!$ pieces.

The next step was to generalize these results to spaces other than the reals.
Abraham, Rubin and Shelah formulated a strengthening of Galvin's theorem for countably based metric spaces of size $\aleph_1$ <cit.>.
Abstracting from many applications, Todorčević introduced the open graph axiom
$\OGA$ <cit.>.
[It is also often called the open coloring axiom $\OCA$.
Note that a graph $G$ on a set $X$ can be identified with a partition or coloring of pairs in $X$, for example by coloring edges in $G$ red and edges in the complement of $G$ blue.
Galvin's theorem is then equivalent to the statement that for any clopen graph $G$ on the reals, there exists either a perfect independent set or a perfect complete subgraph.]
For a topological space $X$,
$\OGA(X)$ states that any open graph $G$ on $X$ either admits a coloring
[Recall that a coloring of a graph $G$ assigns different colors to adjacent vertices.]
in countably many colors or else contains an uncountable complete subgraph.
$\OGA$ states that $\OGA(X)$ holds
for all countably based metric spaces $X$.
$\OGA$ follows from the proper forcing axiom $\mathsf{PFA}$ <cit.>
and has many applications <cit.>.

Feng and Todorčević studied a stronger version of $\OGA(X)$, the open graph dichotomy $\OGD(X)$, that is useful for applications to definable sets of reals <cit.>.

Here the uncountable
complete subgraph
has to be perfect.
$\OGD(X)$ is provable for all analytic sets $X$ of reals and it is in fact consistent that $\OGD(X)$ holds for all definable sets $X$ <cit.>.[See <cit.>*Theorem 1.1 and Section 4, as well as <cit.>*pages 676-677 for an alternative approach for analytic sets due to Todorčević.]
The previous statements only mention open graphs because they cannot hold for all closed graphs <cit.>.
[See Example <ref> in this paper.]

The verbatim analogues of the open graph axiom and dichotomy fail in dimension $3$ and higher:
an open dihypergraph
[A dihypergraph, or directed hypergraph, of dimension $\ddim$ is a set of nonconstant functions with domain $\ddim$.]
on the Baire or Cantor space might neither admit a countable coloring nor have an uncountable complete subhypergraph <cit.>.${}$[See Examples <ref> and <ref> in this paper.]
Keep this todo-note. We added the symbols ${}$ just before the next footnotes 6 and 7 to make the superscript number appear in the right size. It looks like a latex bug.
However, one can replace the latter
by a continuous homomorphic image of a canonical large dihypergraph.
Carroy, Miller and Soukup
proved a countably infinite dimensional version of $\OGD(X)$ for
analytic sets $X$
and extended this to
all sets of reals assuming the axiom of determinacy $\AD$
[It follows that this dichotomy is consistent with $\ZFC$ for all box-open $\aleph_0$-dimensional dihypergraphs on the Baire space which are definable from a countable sequence of ordinals. To see this,  take a model of $\AD+\DC+{V=L(\RR)}$ and add a well-order of the reals by a $\sigma$-closed homogeneous forcing.]
Dorottya's note to self for the footnote: Since the forcing (in the 2nd sentence) is $\sigma$-closed, no new $\omega$-sequences of ordinals are added. Hence by homogeneity, every box-open definable $\omega$-dihypergraph $H$ on the Baire space in the generic extension is in already in $V$ and is definable in $V$. (The proof is similar to Lemma <ref> <ref>. “Definable” means definable from an $\omega$-sequence of ordinals, as usual.) Homogeneity is needed since otherwise the forcing can make any dihypergraph definable even without parameters.
A similar idea
is implicit in work of
Avilés and Todorčević <cit.>.${}$[We would like to thank Stevo Todorčević for pointing out that the idea for a higher dimensional version of $\OGD(X)$ is implicit in the proof of <cit.>.]
The main motivation behind this dichotomy
is to provide new proofs and strong versions of several seemingly unrelated results in descriptive set theory <cit.>.
The first such result is the Hurewicz dichotomy that
characterizes the circumstances in which an analytic subset of a Polish space is contained in a $K_\sigma$ subset
The Hurewicz dichotomy has also been studied by theoretical computer scientists <cit.>.
A dichotomy due to Kechris, Louveau and Woodin describes when
an analytic subset of a Polish space can be separated from a disjoint subset by an $F_\sigma$ set
A celebrated theorem of Jayne and Rogers characterizes $G_\delta$-measurable functions on analytic sets <cit.>.

We study the above problems in the more general context where $\omega$ is replaced by a regular infinite cardinal $\kappa$ with $\kappa^{<\kappa}=\kappa$.
The topological properties of the generalized Baire space ${}^\kappa\kappa$ of functions on $\kappa$ resemble that of the Baire space ${}^\omega\omega$. In fact, larger classes of spaces including ${}^\kappa\kappa$ have been identified as analogues to

Polish spaces
at uncountable cardinals <cit.>.
The study of descriptive set theory for these spaces in the uncountable setting is motivated,

inter alia,
by connections
with model theory <cit.>

and Shelah's classification theory in particular <cit.>.
There is an ample amount of background literature
on generalized Baire spaces and their descriptive set theory,
beginning with the study of closed, $\kappa$-analytic and $\kappa$-coanalytic subsets <cit.>.
It is known that some properties of definable subsets of generalized Baire spaces are closely linked to statements in combinatorial set theory, for instance to the existence of Kurepa trees <cit.>.
The extension of several classical dichotomies to the uncountable setting paves the way for a structure theory of definable sets.
For instance, the Hurewicz dichotomy for $\kappa$-analytic sets is consistent
Assuming the existence of an inaccessible cardinal,
it is consistent for all definable sets <cit.>,
and so is the $\kappa$-perfect set property for definable sets <cit.>.
Moreover, the analogue of the open graph dichotomy for $\kappa$-analytic sets is consistent relative to an inaccessible cardinal <cit.>.
There is keen interest in analogues of the open graph axiom $\OGA$
and similar principles at cardinals beyond $\omega_1$ <cit.>.

The aim here is to extend the aforementioned results on the open graph dichotomy and
its infinite dimensional version
to uncountable cardinals and utilize such results to derive applications to the descriptive set theory of generalized Baire spaces.
We next introduce the relevant variants of these dichotomies.
The precise definitions of some of the following concepts will be given in Section <ref>.
We assume that $\kappa$ is a regular infinite cardinal with
throughout this paper, unless otherwise stated.
The $\kappa$-Baire space
is the set ${}^\kappa\kappa$ of functions from $\kappa$ to $\kappa$ equipped
with the bounded topology (or $\lle\kappa$-box topology),
i.e., the topology generated by the base $\{N_s:s\in{}^{<\kappa}\kappa\}$, where
$N_s=\{x\in{}^\kappa\kappa : s\subseteq x\}$
for each $s\in{}^{<\kappa}\kappa$.
[The $\omega$-Baire space is the Baire space ${}^\omega\omega$, since the bounded topology equals the product topology for $\kappa=\omega$.]
The $\kappa$-Cantor space is ${}^\kappa 2$ with the subspace topology.
A graph is a symmetric irreflexive binary relation.
A graph $G$ on a topological space $X$ is open if it is an open subset of the product space $X\times X$.
A subset of $X$ is $G$-independent if it contains no edges, and
a $\kappa$-coloring of $G$ is a partition of $X$ into at most $\kappa$ many disjoint $G$-independent sets.

open graph axiom
can be generalized to the uncountable setting.
For any class $\mathcal C$, let
$\OGA_\kappa({\mathcal C})$ state that the following holds for all subsets
$X\in \mathcal{C}$ of ${}^\kappa\kappa$:
If $G$ is an open graph on $X$, then
either $G$ admits a $\kappa$-coloring or
$G$ has a complete subgraph of size $\kappa^+$.

The following generalization of
the open graph dichotomy <cit.> is a stronger version of this axiom.

For any class $\mathcal C$,
$\OGD_\kappa({\mathcal C})$ states that the following holds for all subsets $X\in \mathcal{C}$ of ${}^\kappa\kappa$:

If $G$ is an open graph on $X$, then
either $G$ admits a $\kappa$-coloring or
$G$ has a $\kappa$-perfect
complete subgraph.

[I.e., $G$ has a complete subgraph whose domain is a $\kappa$-perfect subset of ${}^\kappa\kappa$.
See Definitions <ref> and <ref>.]

Note that $G$ has a $\kappa$-perfect complete subgraph if and only if there exists
a continuous homomorphism from the complete graph
$\gK{{}^\kappa 2}$ to $G$.[See Corollary <ref>.]
Thus, $\OGD_\kappa(X)$ can be formulated for arbitrary topological spaces as well.
In the countable case,[I.e., for $\kappa=\omega$.]
Carroy, Miller and Soukup introduced the box-open $\ddim$-dimensional dihypergraph dichotomy as an analogue of $\OGD_\omega(X)$ in higher dimensions $\ddim$ <cit.>.
We now consider its analogue for regular infinite cardinals $\kappa$ with $\kappa^{<\kappa}=\kappa$.

Let $X$ be any set and $\ddim$ a set of size at least $2$.
A $\ddim$-dihypergraph[I.e., a $\ddim$-dimensional directed hypergraph.]
on $X$ is a subset $H$ of ${}^\ddim X$ consisting of non-constant sequences.

A subset of $X$ is $H$-independent
if it contains no hyperedges,
[See Subsection <ref> for the precise definitions of the concepts discussed in this paragraph.]
and a $\kappa$-coloring of $H$ is a partition of $X$ into at most $\kappa$ many disjoint $H$-independent sets.
A homomorphism from one $\ddim$-dihypergraph to another is a map that takes hyperedges to hyperedges.
We write elements of ${}^\ddim X$ as $\bar{x}=\langle x_i : i\in\ddim\rangle$.
Let $\dhH\ddim$ denote the following $\ddim$-dihypergraph on ${}^\kappa\ddim$ (see Figure <ref>):

In the next figure, “abovecaptionskip” (distance between figure and caption) and “intextsep” (distance between text and figure) were set LOCALLY. Remove before submitting to journal?

\begin{equation*}
\dhH\ddim
\bigcup_{t\in{}^{<\kappa}\ddim}\prod_{i\in \ddim} N_{t\conc\langle i\rangle}
\{\bar x\in{}^\ddim({}^{\kappa}\ddim)
\::\:
{\exists t\in{}^{<\kappa}\ddim}\ \ {\forall i\in\ddim}\ \
t\conc\langle i\rangle\subseteq x_i
\}
\end{equation*}

(0*,0) edge[-] (-4*,4*)
(0*,0) edge[-] (4*,4*);

(-0.75*,0.75*) edge[-] (-0.4*,1.5*)
(-0.4*,1.5*) edge[-] (-0.6*,2.25*);

(-0.6*,2.25*) edge[thick, -, YellowArrow] (-1.4*,3*)
(-0.6*,2.25*) edge[thick, -, DarkGreenArrow] (-0.6*,3*)
(-0.6*,2.25*) edge[thick, -, TealArrow] (0.2*,3*);

(-1.4*,3*) edge[thick, -, dashed, YellowArrow] (-1.8*,4*)
(-0.6*,3*) edge[thick, -, dashed, DarkGreenArrow] (-0.6*,4*)
(0.2*,3*) edge[thick, -, dashed, TealArrow] (0.8*,4*);

(-0.2*,3*) node  ..
(1.2*,3*) node  ..;

(-1.2*,3*) node[left]  $t_0$
(-0.9*,3*) node  $t_1$
(0.1*,3*) node[right]  $t_\alpha$;

(1.8*,4.2*) node  ..;

(-1.8*,4*) node[above]  $x_0$
(-0.6*,4*) node[above]  $x_1$
(1*,4*) node[above]  $x_\alpha$;

(6*,3*) node $t_\alpha=t^\smallfrown \langle\alpha\rangle$;

(0*,0*) node[above]
(4*,4*) node[above] ;

We will understand $d$ as a discrete topological space and equip ${}^\kappa d$ with the ${<}\kappa$-box topology.
Ordinals are equipped with the discrete topology.
If $X$ is a subset of the $\kappa$-Baire space,
then ${}^\ddim X$ is equipped with the box-topology unless otherwise stated.

For any $\ddim$-dihypergraph $H$

on a topological space $X$,
let $\ODD\kappa H$ denote the following statement:
We define $\ODD\kappa H$ for all dihypergraphs $H$ on $X$ because we sometimes talk about the case when $H$ is not box-open

$\ODD\kappa H$:
Either $H$ admits a $\kappa$-coloring, or
there exists a continuous homomorphism

$f:{}^\kappa\ddim\to X$
from $\dhH{\ddim}$ to $H$.

For classes $\mathcal C$ and $\mathcal D$, let $\ODD\kappa\ddim(\mathcal C,\mathcal D)$ denote statement that
$\ODD\kappa{H\restr X}$ holds
[$H\restr X:=H\cap{}^\ddim X$ denotes the restriction of $H$ to $X$.]
\beginenumerate-(a)

* $X$ is a subset of ${}^\kappa\kappa$ in $\mathcal C$ and

* $H$ is a box-open
[I.e, $H$ is an open subset of ${}^\ddim({}^\kappa\kappa)$ in the box topology.]
$\ddim$-dihypergraph on ${}^\kappa\kappa$ in $\mathcal D$.
\endenumerate-(a)
If $\mathcal D$ is the class of all sets or all $\ddim$-dihypergraphs on ${}^\kappa\kappa$, then we omit it from the notation
and write $\ODD\kappa\ddim({\mathcal C})$ for short.
If $\mathcal C$ has one single element $X$, then we write $X$ instead of $\mathcal C$, and similarly for $\mathcal D$.

For example, $\ODD\kappa\ddim(X)$ states that $\ODD\kappa {H{\restr}X}$ holds for all box-open $\ddim$-dihypergraphs $H$ on ${}^\kappa\kappa$.
It is equivalent that $\ODD\kappa H$ holds for all $\ddim$-dihypergraphs $H$ on $X$ that are box-open on $X$.
[In general, we have to work with the restrictions of dihypergraphs $H\in\mathcal D$ on ${}^\kappa\kappa$ to $X$ instead of dihypergraphs on $X$ which are in $\mathcal D$.
For instance, $\ODD\kappa{H{\restr}X}$ might hold for all definable box-open dihypergraphs $H$
even though $H{\restr}X$ is not necessarily definable.
This is relevant for several applications.]
this is exactly what it's called in <cit.>
Following <cit.>, we call $\ODD\kappa\ddim(X)$ the box-open $\ddim$-dimensional dihypergraph dichotomy for $X$,
or open dihypergraph dichotomy for short.
The open graph dichotomy $\OGD_\kappa(X)$
is equivalent to $\ODD\kappa 2(X)$,
and therefore it
follows from
$\ODD\kappa \ddim(X)$ for any $d$.
[See Subsection <ref> and Lemma <ref>.]

Section <ref> collects basic facts that we will use throughout the paper.
Section <ref> provides tools for working with dihypergraphs and homomorphisms.

Let $\defsetsk$ denote the class of all sets that are definable from a $\kappa$-sequence of ordinals.
The following main result of this paper
is proved in
Section <ref>.

Suppose $\kappa<\lambda$ are regular infinite cardinals and $2\leq\ddim\leq\kappa$.
For any $\Col(\kappa,\lle\lambda)$-generic filter $G$ over $V$, the following statements hold in $V[G]$:
\beginenumerate-(1)

$\ODD\kappa\ddim(\defsets\kappa,\defsets\kappa)$ if $\lambda$ is inaccessible in $V$.
[This is equivalent to (1) in the abstract by
Lemma <ref>.]

$\ODD\kappa\ddim(\defsets\kappa)$ if $\lambda$ is Mahlo in $V$.
[Recall that a cardinal $\lambda$ is Mahlo if the set of inaccessible cardinals $\nu<\lambda$ is stationary in $\lambda$.]
\endenumerate-(1)

A crucial idea in the proof of the uncountable case is to construct a forcing which adds a perfect tree of generic filters over an intermediate model that induces a homomorphism of dihypergraphs.
This technical part of the proof is done in Subsection <ref>.

Note that $\ODD\kappa\ddim(\defsetsk,\defsetsk) \Longrightarrow \ODD\kappa\ddim(\defsetsk)$ holds for any $\ddim<\kappa$, since
any box-open $\ddim$-dihypergraph on ${}^\kappa\kappa$ is in $\defsetsk$ by virtue of the topology having a base of size $\kappa$.
[See Lemma <ref>.]
It thus suffices to assume in <ref> that $\lambda$ is inaccessible in $V$ if $\ddim<\kappa$.
It is open whether the Mahlo cardinal is neccessary for <ref> if $\ddim=\kappa$.
However, the inaccessible cardinal is necessary for <ref>, since the statement implies the $\kappa$-perfect set property $\PSP_\kappa(X)$
[See Definition <ref>. Note that $\PSP_\kappa(X)$ from $\OGD_\kappa(X)$. To see this, consider the complete graph $\gK{X}$ on $X$.]
for all closed sets $X\subseteq{}^\kappa\kappa$ and
thus the existence of an inaccessible cardinal above $\kappa$ in $L$.
[See <cit.>
and <cit.>.
The inaccessible cardinal is also neccessary in the countable case
by <cit.>.]

We also obtain a global version of Theorem <ref> <ref> for all regular infinite cardinals
in Subsection <ref>.
It remains open whether <ref> admits a global version.

In Section <ref>,
we investigate alternatives to the open dihypergraph dichotomy.
We observe
$\ODD\kappa\ddim(X)$ is equivalent to a dichotomy
for $\ddim$-hypergraphs
[See Definition <ref>.]
with the open graph dichotomy $\OGD_\kappa(X)$ as a special case for $\ddim=2$.
[See Subsection <ref>.]
We then show that the definition of $\ODD\kappa\ddim(X)$ is optimal in various aspects.
While $\dhH\ddim$ shows that for any $\ddim\geq 3$, the existence of a continuous homomorphism cannot be replaced by the existence of a large complete subhypergraph,[See Subsection <ref> for stronger counterexamples.]
it is natural to ask for continuous homomorphisms with additional properties such as injectivity.
We study this in Subsection <ref>.
While it cannot be done for $\ddim=\kappa=\omega$,[The dihypergraph in Definition <ref> is a counterexample by Propositions <ref> and <ref>.]
If $\ddim<\kappa$, then we can always choose it to be a homeomorphism onto a closed set (even when $\kappa=\omega$).
it is possible in the uncountable setting
assuming a weak version
of $\Diamond_\kappa$
by the next result.
This follows from Theorems <ref> and <ref>.

Assume $\Diamondi\kappa$.
[$\Diamondi\kappa$ and its relationship with some other versions of $\Diamond_\kappa$ is discussed in Subsection <ref>.
Note that $\Diamondi\kappa$ implies that $\kappa$ is uncountable by Remark <ref>.]
If $H$ is a box-open $\ddim$-dihypergraph on a subset of ${}^\kappa\kappa$ where $2\leq\ddim\leq\kappa$,
then $\ODD\kappa H$ is equivalent to the following statement:

$\ODDI\kappa H$:
Either $H$ admits a $\kappa$-coloring or
there exists an injective continuous homomorphism
from $\dhH{\ddim}$ to $H$.

For $\ddim<\kappa$, or alternatively with additional assumptions on $H$, the continuous homomorphism in $\ODD\kappa H$ can
be chosen to be a homeomorphism onto a closed subset even
without assuming $\Diamondi\kappa$.
[See Theorems <ref>, <ref> and Definition <ref>. This cannot be done in general by  Propositions <ref> and <ref>.]
It is open whether the assumption $\Diamondi\kappa$ in the previous theorem can be removed for $\ddim=\kappa$.
$\Diamondi\kappa$ holds for all inaccessible cardinals $\kappa$ and all successor cardinals $\kappa\geq\omega_2$ with $\kappa^{<\kappa}=\kappa$ by a result of Shelah <cit.>.
[Shelah <cit.> proved that
$\Diamond_\kappa$ is equivalent to $\kappa^{<\kappa}=\kappa$
for successor cardinals $\kappa\geq\omega_2$.
It is easy to show that $\Diamondi\kappa$ follows from $\Diamond_\kappa$ and that it holds at inaccessible cardinals $\kappa$; see Lemma <ref>.]
Moreover, if $\kappa$ is a regular uncountable cardinal, then $\Diamondi\kappa$ holds in $\Col(\kappa,\lle\lambda)$-generic extensions $V[G]$ where $\lambda>\kappa$ is inaccessible.
$\ODD\kappa\ddim$ can be replaced by $\ODDI\kappa\ddim$ in Theorem <ref> when $\kappa$ is uncountable.
[See Corollary <ref>.]

Section <ref> studies a number of applications of the open dihypergraph dichotomy.
We now describe the main results of the section.
The missing definitions will be introduced in the respective subsections and Section <ref>.
The Hurewicz dichotomy for a subset $X$ of a Polish space
states that $X$ is contained in a $K_\sigma$ set
if and only if it does not contain a closed homeomorphic copy
of the Baire space.
Hurewicz showed this for Polish subspaces $X$
[It follows immediately from <cit.> by passing to a compactification that an analytic subset of a Polish space is $K_\sigma$ if and only if does not contain a relatively closed homeomorphic copy of the Baire space, but this statement does not seem to be mentioned directly in the paper. We call this the exact Hurewicz dichotomy (see Section <ref>).]
Saint Raymond <cit.> and Kechris <cit.> proved it for analytic sets.
Kechris extended
it to more complex sets under appropriate determinacy assumptions
In the uncountable setting, the Hurewicz dichotomy has a topological version
similar to the classical one <cit.> as well as a combinatorial version
using superperfect trees <cit.>.
The next result is proved in Theorems <ref> and <ref> using variants of dihypergraphs from <cit.>.

For all subsets $X$ of ${}^\kappa\kappa$,
$\ODD\kappa\kappa(X,\defsetsk)$ implies the following analogues of the Hurewicz dichotomy:
\beginenumerate-(1)

* $\THD_\kappa(X)$:
$X$ can be covered by $\kappa$ many $\kappa$-compact

[A topological space is $\kappa$-compact (or $\kappa$-Lindelöf) if every open cover has a subcover of size strictly less than $\kappa$.]
subsets of ${}^\kappa\kappa$, or
$X$ contains a closed homeomorphic copy of ${}^\kappa\kappa$.

* $\HD_\kappa(X)$:
$X$ can be covered by the sets of branches of at most $\kappa$ many $\lle\kappa$-splitting subtrees of ${}^{<\kappa}\kappa$, or
$X$ contains a $\kappa$-superperfect subset.
[See Definition <ref>.]
\endenumerate-(1)

In fact, we obtain a generalization $\THD_\kappa(X,Y)$ of $\THD_\kappa(X)$ for all supersets $Y$ of $X$ from $\ODD\kappa\kappa(X)$.

[See Definition <ref> and Theorem <ref>. If $Y$ is in $\defsetsk$, then $\ODD\kappa\kappa(X,\defsetsk)$ suffices.]
Here, the $K_\kappa$ set in the first option needs to be a subset of $Y$,
while it suffices that the homeomorphic copy of ${}^\kappa\kappa$ is a relatively closed subset of $Y$ in the second option.

We will use this principle to answer two questions of Zdomskyy in Section <ref>.

Kechris, Louveau and Woodin
proved a dichotomy that characterizes when
an analytic subset of a Polish space
can be separated from a disjoint subset by an $F_\sigma$ set <cit.>.
The special case of complements describes when an analytic set is $F_\sigma$.
This was already shown by Hurewicz <cit.>*Section 6.
The theorem of Kechris, Louveau and Woodin strengthens the Hurewicz dichotomy for analytic sets in the sense that one can derive the latter by a short argument using compactifications.
[See <cit.>.
Note that this argument has a version for weakly compact cardinals by
Proposition <ref> and Corollary <ref>, but it does not generalize to an uncountable regular cardinal that is not weakly compact.]
Let $\RR_\kappa$ denote the set of elements of ${}^\kappa 2$ that take
nonzero values
unboundedly often and $\QQ_\kappa$ the set of those that do not.
The proof of the next theorem is based on ideas from <cit.>.

Suppose that $X$ is a subset of ${}^\kappa\kappa$.
$\ODD\kappa\kappa(X)$ implies the following analogue of the Kechris-Louveau-Woodin dichotomy
for all subsets $Y$ of ${}^\kappa\kappa$ disjoint from $X$:
[If $Y$ is in $\defsetsk$, then $\ODD\kappa\kappa(X,\defsetsk)$ suffices.]

Either there is a $\Fsigma(\kappa)$ set $A$ separating $X$ from $Y$,
[I.e., $X\subseteq A$ and $A\cap Y=\emptyset$.]

or there is a homeomorphism
$f$ from ${}^\kappa 2$ onto a closed subset of ${}^\kappa\kappa$
that reduces
$(\RR_\kappa,\QQ_\kappa)$ to $(X,Y)$.
[I.e., $f(\RR_\kappa)\subseteq X$ and $f(\QQ_\kappa)\subseteq Y$.]

We extend the previous theorem to sets $X$ and $Y$ that are not neccessarily disjoint
in Theorem <ref>.
Here, the first option is replaced by the statement
that $X$ is the a union of $\kappa$-many sets $X_\alpha$ with no limit points in $Y$.[See Definition <ref>.
Note that  by Lemma <ref>, this statement is equivalent to the conjunction of the statements that $|X\cap Y|\leq\kappa$ and $X\setminus Y$ can be separated from $Y$ by a $\Fsigma(\kappa)$ set.]
We further derive a more general dichotomy $\KLW_\kappa^H(X,Y)$
that is defined
relative to an arbitrary box-open $\ddim$-dihypergraph $H$ where $2\leq\ddim\leq\kappa$
in Theorem <ref>.
In the first option of the dichotomy, the role of limit points is replaced by a notion of
limit points relative to $H$.[See Definition <ref>.]
In the second option, $\RR_\kappa$ and $\QQ_\kappa$ are replaced by analogous subsets $\RR^\ddim_\kappa$ and $\QQ^\ddim_\kappa$ of ${}^\kappa\ddim$ and moreover,
the map needs to be a continuous homomorphism from $\dhH \ddim$ to $H$ instead of a homeomorphism from ${}^\kappa 2$ onto a closed set.
[See Definition <ref> and the preceding paragraph. Note that in the second option of $\KLW_\kappa(X,Y)$, it is equivalent to ask that $f$ is continuous and injective instead of a homeomorphism by Theorem <ref>. This shows that $\KLW_\kappa(X,Y)$ is equivalent to the special case of $\KLW_\kappa^{\gK{{}^\kappa \kappa}}(X,Y)$ for the complete graph $\gK{{}^\kappa \kappa}$ on ${}^\kappa\kappa$ by Remark <ref>.]
$\KLW^\ddim_\kappa(X)$ then states that $\KLW_\kappa^H(X,Y)$ holds for all subsets $Y$ of ${}^\kappa\kappa$
and all box-open $\ddim$-dihypergraphs $H$ on ${}^\kappa\kappa$.
The next result follows from Theorem <ref> and
Corollaries <ref> and <ref>,
as does its restriction to definable dihypergraphs $H$
and definable subsets $Y$.

The following hold for all $2\leq\ddim\leq\kappa$ and all subsets $X$ of ${}^\kappa\kappa$.
\beginenumerate-(1)

* $\ODD\kappa\kappa(X) \Longrightarrow
\KLW_\kappa^\ddim(X)$.

* $\KLW_\kappa^\ddim(X) \Longrightarrow \ODD\kappa\ddim(X)$.
\endenumerate-(1)
In particular,
is equivalent to

This general version of the Kechris-Louveau-Woodin dichotomy
is used
to prove the next result.
It follows from Lemma <ref> and
Corollary <ref>.

\beginenumerate-(1)

* $\ODD\kappa\kappa({}^\kappa\kappa)\Longrightarrow
\ODD\kappa\kappa(\analytic(\kappa))$.

* $\ODD\kappa\kappa({}^\kappa\kappa,\defsetsk)\Longrightarrow
\ODD\kappa\kappa(\analytic(\kappa),\defsetsk)$.
\endenumerate-(1)

Note that it is easy to see that $\ODD\kappa\kappa({}^\kappa\kappa) \Longrightarrow\ODD\kappa\kappa(X)$ for continuous images $X$ of ${}^\kappa\kappa$,[See Lemma <ref>.]
but not all closed subsets of ${}^\kappa\kappa$ are of this form <cit.>.
It follows from the previous theorem
that $\ODD\kappa\kappa({}^\kappa\kappa,\defsetsk)$ has the same consistency strength as an inaccessible cardinal.

$\ODD\kappa\kappa({}^\kappa\kappa)$ has further applications to Väänänen's perfect set game that is described next.
Any set of reals can be decomposed as the disjoint union of a crowded set and a countable scattered set by Cantor-Bendixson analysis.
[Recall that a set of reals
$X$ is crowded if it has no isolated points, perfect if it is closed and crowded, and scattered if each nonempty subspace contains an isolated point.]
Väänänen showed that an analogue of this statement for all closed subsets of ${}^\kappa\kappa$ is consistent
[Väänänen used a measurable cardinal
and Galgon reduced this to an inaccessible cardinal <cit.>.
In fact, the analogue for closed sets is equivalent to the $\kappa$-perfect set property $\PSP_\kappa(X)$ for closed sets <cit.>.]
but not provable <cit.>.
To this end, he generalized the notions of scattered and crowded sets
to subsets $X$ of ${}^\kappa\kappa$
via a game $\GV_\kappa(X,x)$ of length $\kappa$.
[See Definition <ref>.
Here, $x\in X$ denotes the first move of player $\ptwo$; this is fixed. We also consider a global version $\GV_\kappa(X)$ of the game, where player $\ptwo$ may play any element of $X$ as their first move.]
We extend Väänänen's result to all subsets of ${}^\kappa\kappa$ in the next theorem, which follows from
Theorems <ref> and <ref>.

$\ODD\kappa\kappa({}^\kappa\kappa)$ implies the following version of the Cantor-Bendixson decomposition
for all subsets $X$ of ${}^\kappa\kappa$:
[$\ODD\kappa\kappa({}^\kappa\kappa,\defsetsk)$ suffices if $X$ is in $\defsetsk$.]

$X$ equals the disjoint union of a $\kappa$-scattered set of size $\kappa$ and a $\kappa$-crowded set.
[See Definition <ref>.]

In particular,
Väänanen's perfect set game $\GV_\kappa(X,x)$
is determined for all subsets $X$ of ${}^\kappa\kappa$ and all $x\in X$.
[It is easy to see from the definitions that this statement follows from $\cbii(X)$. Note that this statement implies the determinacy of the global version $\GV_\kappa(X)$ of Väänanen's game as well.]

While there is a verbatim analogue of the Baire property at uncountable cardinals, it is not as useful as in the countable setting since even simple sets such as the club filter do not satisfy it <cit.>.
${}$[See the beginning of Subsection <ref>. Note that the club filter can be coded as a $\kappa$-analytic subset of the $\kappa$-Baire space.]
Footnote size problem fixed in the usual way
The asymmetric $\kappa$-Baire property $\ABP_\kappa(X)$ introduced in <cit.> is a more general principle
that is equivalent to the determinacy of the Banach-Mazur game of length $\kappa$ for $X$.
In the countable setting, $\ABP_\omega(\defsets\omega)$
[I.e., $\ABP_\omega(X)$ for all subsets $X\in\defsets\omega$ of ${}^\omega\omega$.]
is thus equivalent to the Baire property for sets in $\defsets\omega$.
We wrote $\defsets\omega$ instead of $X$ here since. The equivalence only holds for sufficiently closed classes of subsets of ${}^\omega\omega$.
The next result is proved in Theorem <ref>.

For all subsets $X$ of ${}^\kappa\kappa$,
$\ODD\kappa\kappa(X,\defsetsk)$ implies the asymmetric $\kappa$-Baire property $\ABP_\kappa(X)$.

A new consequence in the countable setting is that
$\ODD\omega\omega(\defsets\omega,\defsets\omega)$ implies the Baire property for subsets of the Baire space in $\defsets\omega$.

A celebrated result of Jayne and Rogers characterizes ${\bf \Delta}^0_2$-measurable functions on analytic sets
as those that are $\sigma$-continuous with closed pieces <cit.>.
[A function between countably based metric spaces is ${\bf \Delta}^0_2$-measurable if and only if it is ${\bf \Pi}^0_2$-measurable.
It is $\sigma$-continuous with closed pieces if it can be obtained as a countable union of continuous functions on relatively closed subsets of its domain.]
A simpler proof of this theorem was subsequently discovered by Motto Ros and Semmes <cit.>.
Possible generalizations thereof are a subject of intense study <cit.>.
Recently, Carroy, Miller and Soukup found a new proof that derives the theorem from the open dihypergraph dihotomy <cit.>.
Their proof allows us to generalize the Jayne-Rogers theorem to the uncountable setting.
The next result is proved in Corollary <ref>.

$\ODD\kappa\kappa(\defsetsk,\defsetsk)$ implies the following analogue of the Jayne-Rogers theorem for all subsets $X\in\defsetsk$ of ${}^\kappa\kappa$:

Any function $f:X\to{}^\kappa\kappa$
is $\DeltaZeroTwo(\kappa)$-measurable if and only if it is a union of $\kappa$ many continuous functions on relatively closed subsets of $X$.

At the end of Section <ref>,
we prove some implications and separations
between applications of the open dihypergraph dichotomy such as
the Kechris-Louveau-Woodin dichotomy, its generalization to arbitrary sets,
the $\kappa$-perfect set property, versions of the Hurewicz dichotomy
and the analogue of the Jayne-Rogers theorem.
These rely on the comparison relation between dihypergraphs introduced in Subsection <ref>.
For example, the next results follow as special cases from
Remark <ref>,
Proposition <ref>,

Corollaries <ref> and <ref>.

\beginenumerate-(1)

[$\KLW_\kappa(\mathcal C,\mathcal D)$ states that $\KLW_\kappa(X,Y)$ holds for all subsets $X\in\mathcal C,\,Y\in\mathcal D$ of ${}^\kappa\kappa$. $\THD_\kappa(\mathcal C,\mathcal D)$, $\PSP_\kappa(\mathcal C)$ and $\JR_\kappa(\mathcal C)$ are defined similarly; see Definitions <ref>, <ref> and <ref>.]
[Note that if $\kappa$ is not weakly compact, then

by <cit.>; see <cit.>*Proposition 2.7 or Remark <ref>.
If $\kappa$ is weakly compact, then we have equivalence in <ref> for the class of definable subsets of the $\kappa$-Cantor space by Corollary <ref>.]

$\THD_\kappa(\defsetsk,\defsetsk)\Longrightarrow\JR_\kappa(\defsetsk)$ if $\kappa$ is weakly compact.
\endenumerate-(1)

The above versions of the Hurewicz dichotomy,
the asymmetric $\kappa$-Baire property and the Jayne-Rogers theorem for definable subsets of ${}^\kappa\kappa$
are consistent relative to an inaccessible cardinal by
Theorem <ref>.
The first two statements provide new proofs of the main results of <cit.> and answer two questions of Zdomskyy <cit.>.[See <cit.>.]
The variant $\KLW_\kappa(\defsets\kappa):=\KLW_\kappa(\defsetsk,\powerset({}^\kappa\kappa))$ of the Kechris-Louveau-Woodin dichotomy,

its extension $\KLW^\ddim_\kappa(\defsetsk)$
relative to box-open $\ddim$-dihypergraphs for $2\leq\ddim\leq\kappa$,
the determinacy of Väänänen's perfect set game and
in fact $\cbii(X)$
for arbitrary subsets $X$ of ${}^\kappa\kappa$
are consistent relative to a Mahlo cardinal.
The former answers a question of Lücke, Motto Ros and the first-listed author and solves a recent problem of Bergfalk, Chodounský, Guzmán and Hrušák.
The latter extends a result of Väänänen <cit.> from closed sets to arbitrary subsets of ${}^\kappa\kappa$.
Moreover, it lowers the upper bound for the consistency strength of a dichotomy studied in <cit.>
from a measurable to a Mahlo cardinal.

§ PRELIMINARIES

§.§ Notation

$\alpha$, $\beta$, $\gamma$, $\delta$ denote ordinals and $\kappa$, $\lambda$, $\mu$, $\nu$ cardinals.
$\kappa$ always denotes an infinite cardinal with $\kappa^{<\kappa}=\kappa$, unless otherwise stated.
$\Lim$ and $\Succ$ denote the classes of limit and successor ordinals, respectively.
If $\alpha=\beta+1$, then $\alpha-1$ denotes $\beta$.

For any set $\ddim$,
a $\ddim$-sequence is a function $f: \ddim\to V$.
We write $\ddim$-sequences as $\bar{x}=\langle x_i : i\in\ddim\rangle$.
$\langle c\rangle^{\ddim}$ denotes the constant
$\ddim$-sequence with value $c$.
We write $\bar x^\pi:=\langle x_{\pi(i)}:i\in\ddim\rangle$ for any $D$-sequence $\bar x=\langle x_i:i\in D\rangle$ and $\pi\colon d\rightarrow D$, where $D$ is any set.
A subsequence of $\bar x=\langle x_i:i\in \alpha\rangle$ is $\bar x^\pi=\langle x_{\pi(i)}:i\in \beta\rangle$ for some $\beta\leq\alpha$ and some strictly increasing map $\pi:\beta\to\alpha$.
For any set $\ddim$,
$\alpha<\kappa$ and any sequence
$\bar x=\langle x_i:i\in\ddim \rangle\in{}^\ddim{({}^\kappa\kappa)}$,
let $\bar x\cwrestr\alpha:=\langle x_i\restr\alpha:i\in\ddim\rangle$.
$s\conc t$ denotes the concatenation of two sequences $s$ and $t$,
and $\bigoplus_{\alpha<\beta} s_\alpha$ denotes the concatenation of the sequences
$s_\alpha$ for $\alpha<\beta$.
Given a function $f:X\to Y$, write $f(U)$ for the pointwise image of a subset $U$ of $X$ under $f$, and
write $f^{-1}(W)$ for the preimage of a subset $W$ of $Y$.
For any set $\ddim$, let $f^\ddim:{}^\ddim X\to{}^\ddim Y$ denote the function defined by letting
$f^\ddim(\langle x_i:i\in \ddim\rangle) :=
\langle f(x_i):i\in\ddim\rangle$
for all $\langle x_i:i\in\ddim\rangle\in{}^\ddim X$.
$\id_X$ denotes the identity function on $X$.

A class $X$ is definable from a set $y$ if $X=\{x : \varphi(x,y)\}$ for some first order formula $\varphi(v_0,v_1)$ with two free variables.
Let $\defsets\kappa$ denote the class of those sets which are definable from a $\kappa$-sequence of ordinals.
By a definable set, we always mean an element of $\defsetsk$ when $\kappa$ is clear from the context.
We use the following notation for definable subsets of the $\kappa$-Baire space:
If $\varphi(v_0,v_1)$ is a first order formula with two free variables and $y$ is a set, write
$X^\kappa_{\varphi,y}:=\{x\in{}^{\kappa}\kappa: \varphi(x,y)\}$.
We further write $X_{\varphi,y}$ for $X^\kappa_{\varphi,y}$ if $\kappa$ is clear from the context.

Basic open subsets of ${}^\kappa\ddim$ are denoted
$N_t:=\{x\in {}^\kappa\ddim : t\subseteq x\}$
for any discrete topological space $\ddim$
and any $t\in {}^{<\kappa}\ddim$.
Ordinals are always equipped with the discrete topology.
Suppose that $X$ is a topological space.
The closure of a subset $Y$ of $X$ is denoted by $\closure Y$.
$Y'$ denotes the set of limit points of $Y$,
i.e., the set of those $x\in X$
with $x\in\closure{Y\setminus\{x\}}$.
The $\kappa$-Borel subsets of $X$ are defined by closing the set of basic open sets under unions of length $\kappa$ and complements.
The first level of the $\kappa$-Borel hierarchy is the class ${\bf\Sigma}^0_1(\kappa)$ of open sets.
For any $\gamma$ with $1<\gamma<\kappa^+$, ${\bf\Sigma}^0_\gamma(\kappa)$ sets are of the form $\bigcup_{\alpha<\kappa} A_\alpha$, where each $A_\alpha$ is in ${\bf\Pi}^0_\beta(\kappa)$ for some $\beta<\gamma$.
For any $\gamma$ with $0<\gamma<\kappa^+$, ${\bf\Pi}^0_\gamma(\kappa)$ sets are complements of ${\bf\Sigma}^0_\gamma(\kappa)$ sets, and
a set is ${\bf\Delta}^0_\gamma(\kappa)$ if it is both ${\bf\Sigma}^0_\gamma(\kappa)$ and ${\bf\Pi}^0_\gamma(\kappa)$.

The $\kappa$-analytic or $\analytic(\kappa)$ subsets of ${}^\kappa\kappa$
are subsets of the form $f(Y)$, where $f: {}^\kappa\kappa\to {}^\kappa\kappa$ is continuous and $Y$ is a closed subset of ${}^\kappa\kappa$.
A $\coanalytic(\kappa)$ set is a complement of a $\analytic(\kappa)$ set.
A function $f: X\to Y$ between topological spaces is called $\mathcal{C}$-measurable
with respect to a collection $\mathcal C$ of subsets of $X$
if $f^{-1}(U) \in\mathcal{C}$ for every open subset $U$ of $Y$.

A topological space is $\kappa$-compact (or $\kappa$-Lindelöf) if every open cover has a subcover of size strictly less than $\kappa$.
It is $K_\kappa$ if it is a union of $\kappa$ many $\kappa$-compact subspaces.
A subset $X$ of ${}^\kappa\kappa$ is $\kappa$-precompact if its closure is $\kappa$-compact.

It is bounded by
if $x(\alpha)<b(\alpha)$ for all $x\in X$ and $\alpha<\kappa$.
Write $x\leq^* y$ if there is some $\alpha<\kappa$ such that $x(\beta)<y(\beta)$ for all $\alpha<\beta<\kappa$.
A subset $X$ of ${}^\kappa\kappa$ is eventually bounded by $b\in{}^\kappa\kappa$ if $x\leq^* b$ for all $x\in X$.
An uncountable cardinal $\kappa$ is weakly compact if and only if
${}^\kappa 2$ is $\kappa$-compact if and only if
every closed and bounded subset of ${}^\kappa\kappa$ is $\kappa$-compact.
[See <cit.> and
<cit.>*Lemma 2.6. Conversely, it is easy to see that $\kappa$-compact subsets of ${}^\kappa\kappa$ are always closed and bounded for all infinite cardinals $\kappa$ with $\kappa^{<\kappa}=\kappa$.]

§.§ Dihypergraphs

In this subsection, $X$ denotes a set and $\ddim$ a set of size at least $2$.
$\dhC\ddim{X}$ denotes the diagonal, i.e., the set of constant sequences
$\langle c\rangle^{\ddim}$ with $c\in X$.
Let $\Sym(\ddim)$ denote the set of all permutations of $\ddim$.

\beginenumerate-(a)

* A $\ddim$-dihypergraph[This is short for $\ddim$-dimensional directed hypergraph.] on $X$
is a subset $H$ of ${}^\ddim X\,\setminus\,\dhC\ddim{X}$.
Elements of $H$ are called hyperedges of $H$.

* A $\ddim$-hypergraph on $X$ is a $\ddim$-dihypergraph $H$ on $X$ that is closed under permutation of hyperedges, i.e., $\bar x^\pi\in H$ for all $\bar x\in H$ and all $\pi\in \Sym(\ddim)$.
\endenumerate-(a)

A digraph on $X$ is then a $2$-dihypergraph on $X$, while a graph on $X$ is a $2$-hypergraph on $X$, i.e., a symmetric irreflexive binary relation on $X$.

\beginenumerate-(a)

* $\dhK\ddim X:=
{}^\ddim X-\dhC\ddim{X}$
is the complete $\ddim$-hypergraph on $X$.

* $\gK X:=\dhK 2 X$ is the complete graph on $X$.

* $\dhII\ddim X$ denotes the $\ddim$-dihypergraph consisting of all
injective elements of ${}^\ddim X$.
\endenumerate-(a)

Note that $\dhII 2 X=\gK X$, but $\dhII\ddim X\subsetneq \dhK\ddim X$ if $\ddim\geq 3$ and $|X|\geq 2$.

A graph $G$ on $X$ has a complete subgraph with property $\varphi$ if there exists a subset $Y\subseteq X$ such that $\varphi(Y)$ holds and the complete graph $\gK Y$ on $Y$ is a subset of $G$.
we say that
a $\ddim$-dihypergraph $H$ on $X$
has a complete subhypergraph with property $\varphi$ if there exists a subset $Y\subseteq X$ such that $\varphi(Y)$ holds and $\dhK \ddim Y\subseteq H$.

Let $H$ and $I$ be $\ddim$-dihypergraphs on $X$ and $Y$, respectively. Let $Z$ be a subset of $X$.
\beginenumerate-(a)

* The restriction of $H$ to $Z$ is $H\restr Z:=H\cap{}^\ddim Z$.

* $Z$ is $H$-independent if $H\restr Z=\emptyset$.

* $Z$ is an $H$-clique if $\dhK \ddim Z \subseteq H$.

* Let $\lambda$ be a cardinal. A function $c:X\to\lambda$
is a $\lambda$-coloring of $H$
if $c^{-1}(\{\alpha\})$ is $H$-independent for all $\alpha<\lambda$.

* A homomorphism from $H$ to $I$ is a function $f:X\to Y$
such that $f^\ddim(H)\subseteq I$.
\endenumerate-(a)

Given a family of topological spaces $\langle X_i : i\in \ddim\rangle$,
the box-topology
on $\prod_{i\in \ddim} X_i$ is the topology generated by sets of the form
$\prod_{i\in \ddim} U_i$, where $U_i$ is an arbitrary open subset of $X_i$ for each $i\in \ddim$.
If $X$ is a subset of ${}^\kappa\kappa$, then
we always work with the box topology on ${}^\ddim X$ unless otherwise stated.
We sometimes also consider the product topology, but this is always explicitly stated, and “product-open” etc are defined in footnotes, so we don't need to mention it here.

In the following, let $H$ be a $\ddim$-dihypergraph on a topological space $X$.
We call $H$
box-open on $X$ if it is open as a subset of
${}^\ddim X$ with the box-topology.
For finite $\ddim$, we will simply say that $H$ is open on $X$.

\beginenumerate-(a)

* The domain $\domh H$ of $H$ is the set of all $x\in X$ such that $x$ is an element of some hyperedge of $H$.
[More precisely, $\domh H=\bigcup_{i\in\ddim} \proj_i(H)$, where $\proj_i$ denotes projection onto the $i^\textrm{th}$ coordinate.]

* $H$ is relatively box-open if it is a box-open dihypergraph on its domain $\dom(H)$.
\endenumerate-(a)

Note that $H$ is box-open on $X$ if and only if $H$ is relatively box-open and $\domh H$ is a relatively open subset of $X$.

§.§ Trees and order preserving maps

Let $\ddim$ and $D$ denote discrete topological spaces of size at least $2$ throughout this subsection.

Suppose $s,t\in{}^{\leq\kappa}\ddim$ and $A\subseteq{}^{\leq\kappa}\ddim$.
Let $\lh(t):=\dom(t)$ and $\lh(A):=\sup\{\lh(u):u\in A\}$.
Let $\kkppred t:=\{s\in{}^{<\kappa}\ddim: s\subsetneq t\}$
and $\succ{t}:=\{u\in {}^{<\kappa}\ddim: t\subsetneq u\}$.
$s$ and $t$ are called compatible,
denoted $s \compat t$, if $s\subseteq t$ or $t\subseteq s$.
[For general posets $\PP$, this is called comparable. Conditions $p,q\in \PP$ are called compatible if there exists a common extension $r\leq p,q$.]
$s$ and $t$ are called incompatible, denoted $s \perp t$, if they are not compatible.
Let $s\wedge t$ denote the maximal $r\in {}^{\leq\kappa}\ddim$ with $r\subseteq s$ and $r\subseteq t$.
Note that $s\wedge t$ is the node where $s$ and $t$ split if $s\perp t$, and $s\wedge t=s$ if $s\subseteq t$.
For incompatible $s$ and $t$, let $\Delta(s,t):=\min\{ \alpha<\kappa : s(\alpha)\neq t(\alpha)\}$.
Then $\Delta(s,t)=\lh(s\wedge t)$ and $s\wedge t= s\restr\Delta(s,t)=t\restr\Delta(s,t)$.

A subtree of ${}^{<\kappa}\ddim$ is a subset $T$ that is closed under forming initial segments (i.e., it is downwards closed).
$[T]:=\{ x\in {}^\kappa\ddim :\: {\forall \alpha<\kappa}\ \ {x{\restr}\alpha \in T}\}$ denotes its set of branches.
Note that it is a closed subset of ${}^\kappa\ddim$.
The height of $T$ is defined as $\height(T):=\sup\{\lh(t)+1:t\in T\}$.
Thus, $\height(T)=\lh(T)$ if $\height(T)\in\Lim\cup \{0\}$ and $\height(T)=\lh(T)+1$ if $\height(T)\in\Succ$.
The stem of $T$ is
the unique maximal node $s\in T$ with $s\compat t$ for all $t\in T$.

If $A$ is a subset of ${}^{\leq\kappa}\ddim$, let
$T(A):=\{t\in{}^{<\kappa}\ddim :\: {\exists a\in A}\ \ {t\subseteq a} \}$
$T(A)$ is used for $A\subseteq{}^{<\kappa}\ddim$ in Remark <ref> and Def <ref>
denote the tree of initial segments of its elements.
Note that if $X\subseteq{}^\kappa\ddim$,
$[T(X)]$ is the closure of $X$.

Suppose $T$ is a subtree of ${}^{<\kappa}\ddim$.
\beginenumerate-(a)

* $T$ is pruned if every node $t\in T$ extends to a branch $x\in [T]$.

* A node $t$ in $T$ is splitting if it has at least two direct successors in $T$.

* $T$ is cofinally splitting if for each $t\in T$, there exists a splitting node $u$ in $T$ with $t\subseteq u$.

* $T$ is ${<}\kappa$-closed if for any strictly increasing sequence $\bar{t}=\langle t_i : i<\alpha \rangle$ in $T$ with $\alpha<\kappa$, there exists an upper bound $t\in T$ for $\bar{t}$.

* $T$ is $\kappa$-perfect if it is cofinally splitting and ${<}\kappa$-closed.

* A closed subset $X$ of ${}^\kappa\ddim$ is $\kappa$-perfect if $T(X)$ is $\kappa$-perfect.
\endenumerate-(a)
$\kappa$-perfect set property
holds for a subset $X$ of ${}^\kappa\ddim$ if either $|X|\leq\kappa$ or $X$ contains a $\kappa$-perfect subset.
$\PSP_\kappa(\mathcal C)$ states that
We need to keep ${}^\kappa\kappa$ instead of ${}^\kappa\ddim$ in the definition of $\PSP_\kappa(\mathcal C)$
holds for all subsets $X\in \mathcal C$ of ${}^\kappa\kappa$.

Suppose that $\PP$ and $\QQ$ are posets.
A partial function $\iota: \PP\partialto \QQ$
is called strict order preserving if  $p < q$ implies $\iota(p)< \iota(q)$ for all
$\iota$ is called strict order reversing if  $p < q$ implies $\iota(p) > \iota(q)$ for all

Let $\iota$ be a partial function from ${}^{<\kappa}\ddim$ to ${}^{<\kappa}D$.
\beginenumerate-(a)

$\iota$ is $\perp$-preserving if $s\perp t$ implies $\iota(s)\perp\iota(t)$ for all $s,t\in\dom(\iota)$.

Suppose that $\iota$ is strict order preserving.
$\iota$ is called continuous
if $\iota(t)=\bigcup_{s\subsetneq t}\iota(s)$
for all $t\in\dom(\iota)$ with  $\lh(t)\in\Lim$
and $\kkppred t\subseteq\dom(\iota)$.
\endenumerate-(a)

Let $\iota$ be a strict order preserving partial function from ${}^{<\kappa}\ddim$ to ${}^{<\kappa}D$.
\beginenumerate-(a)

inserted pagebreak here
$[\iota]$ denotes the partial function from ${}^\kappa\ddim$ to ${}^\kappa D$
$\dom([\iota])$ consists of those
This more complicated def of $\dom([\iota])$ is needed since we won't always assume that $\dom(\iota)$ is a subtree
$x\in{}^{\kappa}\ddim$ with $x\restr\alpha\in\dom(\iota)$  for unboundedly many  $\alpha<\kappa$, and for all $x\in\dom([\iota])$

\[
\bigcup_{\substack{\alpha<\kappa,\\ x\restr\alpha\in\dom(\iota)}} \iota(x\restr\alpha).
%\bigcup\{\iota(x\restr\alpha):\:\alpha<\kappa,\ x\restr\alpha\in\dom(\iota)\}.
\]

denote the tree of initial segments of elements of $\ran(\iota)$.
\endenumerate-(a)

We will usually assume that $\dom(\iota)$ is a pruned subtree of ${}^{<\kappa}\ddim$,
For $\dom([\iota])=[\dom(\iota)]$, it suffices that $\dom(\iota)$ is a subtree. For $T(\iota)=T(\ran([\iota]))$,
it suffices that every $s\in\dom(\iota)$ extends to some $x\in\dom([\iota])$ (but $\dom(\iota)$ need not be a subtree).
in which case $\dom([\iota])=[\dom(\iota)]$ and
$\iota(x)=\bigcup_{\alpha<\kappa} \iota(x\restr\alpha)$ for all $x\in\dom([\iota])$.
$T(\iota)=T(\ran([\iota]))$ in this case,
and therefore
$[T(\iota)]$ is the closure of $\ran([\iota])$ in ${}^\kappa D$.

Let $\iota$ be a strict order preserving partial function from ${}^{<\kappa}\ddim$ to ${}^{<\kappa} D$.
\beginenumerate-(1)

$[\iota]$ is a continuous function from
to ${}^\kappa D$.

If $\iota$ is $\perp$-preserving,
$[\iota]$ is a homeomorphism from $\dom([\iota])$ to $\ran([\iota])$.
\endenumerate-(1)

Since $[\iota]$ is strict order preserving, we have
$[\iota]\big(N_t\cap \dom([\iota])\big)\subseteq N_{\iota(t)}$
for all $t\in\dom(\iota)$.
<ref> follows easily from this observation.
To show <ref>,
suppose that $\iota$ is $\perp$-preserving.
Then $N_{\iota(s)}\cap N_{\iota(t)}=\emptyset$
for all $s,t\in\dom(\iota)$ such that $s\perp t$.
Therefore $[\iota]$ is injective and
$[\iota]\big(N_t\cap \dom([\iota])\big)= N_{\iota(t)}\cap\ran([\iota])$
for all $t\in\dom(\iota)$.
This implies that $[\iota]$ is a homeomorphism from $\dom([\iota])$ to $\ran([\iota])$.

Note that the converse of <ref> fails for the map
$\iota:{}^{<\kappa}\kappa\to{}^{<\kappa}\kappa$ defined by fixing a nonempty element $s$ of ${}^{<\kappa}\ddim$ and
$\iota(t):= s$ for all $t\in{}^{<\kappa}\ddim$ with $\lh(t)=1$
$\iota(t)=s\conc t$ for all $t\in{}^{<\kappa}\ddim$ with $\lh(t)>1$.

The converse of <ref> does hold for maps $\iota$ such that $\ran([\iota])\cap N_{\iota(t)}=[\iota](N_t)$ for all $t\in\dom(\iota)$.
In fact, if $\iota$ has the latter property and
$[\iota]$ is injective, then $\iota$ is $\perp$-preserving.

A $\wedge$-semilattice is a
poset $\PP$ such that for any two $p,q\in \PP$, there exists a greatest lower bound $p \wedge q$.
We regard
${}^{<\kappa}\ddim$ and ${}^{<\kappa} D$ as a $\wedge$-semilattices with respect to $\subseteq$.
[The greatest lower bound $s\wedge t$ of $s,\,t$ of ${}^{<\kappa}\ddim$ is defined at the beginning of this subsection.]

Suppose $\PP$ and $\QQ$ are $\wedge$-semilattices and $\iota:\PP\to\QQ$.
\beginenumerate-(i)

* $\iota$ is a $\wedge$-homomorphism if $\iota(p\wedge q)=\iota(p)\wedge \iota(q)$ for all $p,q\in \PP$.

* $\iota$ is a strict $\wedge$-homomorphism if it is a strict order preserving $\wedge$-homomorphism.
\endenumerate-(i)

Note that any $\wedge$-homomorphism is order preserving,
but not necessarily strict order preserving.

Moreover, a strict order preserving map
“order preserving” does not suffice for the right-to-left-direction. To see this, let $\iota(\emptyset)=\emptyset$,
and for all $s\in{}^{<\kappa}\kappa$, let
$\iota(\langle 0\rangle\conc s)=s$ and $\iota(\langle\alpha\rangle\conc s)=\langle\alpha\rangle\conc s$ if $\alpha\geq 1$.
$\iota: {}^{<\kappa}\ddim\to {}^{<\kappa} D$ is a $\wedge$-homomorphism if and only if
$\iota(t\conc\langle i\rangle) \wedge
\iota(t\conc\langle j\rangle)
= \iota(t)$
for all $t\in{}^{<\kappa}\ddim$ and all $i,j\in\ddim$ with $i\neq j$.[Note that if $\iota: {}^{<\kappa}\ddim\to {}^{<\kappa}D$ is strict order preserving, then $\iota$ is a strict $\wedge$-homomorphism with $\lh(\iota(t))=\lh(t)$ for all $t\in{}^\kappa\kappa$ if and only if $[\iota]$ is an isometry in the standard $\kappa$-ultrametric.]
This is a strong form of $\perp$-preservation.[$\perp$ refers to reverse inclusion on ${}^{\leq\kappa}\ddim$ throughout the paper.]

§.§ Forcing

A forcing $\PP=(P,\leq_\PP,\one_\PP)$ is a triple such that $\leq_\PP$ is a pre-order (i.e., a reflexive transitive relation) on $P$ and $\one_\PP$ is a largest element of $\PP$ with respect to $\leq_\PP$. We confuse $\PP$ with its domain $P$, and we usually write $\leq$, $\perp$, $\one$
and $\forces$ instead of $\leq_{\PP}$, $\perp_\PP$, $\one_\PP$ and $\forces_\PP$, respectively.
For all $p,q\in \PP$, we let $p\compat q$ (or sometimes $p\compat_\PP q$) denote the statement that $p$ and $q$ are compatible, i.e., that $p\not\perp q$.
Let $\BB(\PP)$ denote the Boolean completion of $\PP$.[Note that $\BB(\PP)$ is unique up to isomorphism.]
If $\PP$ is separative, we assume that $\PP$ is a dense subset of $\BB(\PP)$.

\beginenumerate-(a)

* An atom in a forcing $\PP$ is a condition $p\in\PP$ with no incompatible extensions. A forcing $\PP$ is non-atomic if it has no atoms.

* A forcing $\PP$ is homogeneous if for all $p,q\in \PP$, there is an automorphism $\pi: \PP\to \PP$ such that $\pi(p)$ and $q$ are compatible.
\endenumerate-(a)

For separative posets, $i:\PP\to\QQ$ is a dense embedding iff it's a sub-isomorphism (as defined in Philipp's paper).

Suppose $\PP,\QQ$ are forcings.
\beginenumerate-(a)

* A dense embedding $\iota:\PP\to\QQ$ is a homomorphism with respect to $\leq,\,\perp$ and $\one$
such that $\iota(\PP)$ is a dense subset of $\QQ$.

Two forcings $\PP$ and $\QQ$ are
($\PP\simeq \QQ$) if there exist
dense embeddings
$\iota: \PP\to \RR$ and $\nu: \QQ\to\RR$ into some forcing $\RR$.

Let $\iota: \PP\to \QQ$ be a dense embedding. Define a $\PP$-name
$\sigma^\iota$ for each $\QQ$-name $\sigma$ by recursion on the rank as

\[
\sigma^{\iota}:=\left\{(\tau^{\iota},p) \::\:
p\in \PP,\ \,
\exists q\in\QQ\ \big(\iota(p)\leq q\wedge (\sigma,q)\in \tau\big)
\right\}.
\]

\endenumerate-(a)

Recall the following standard facts.[See e.g. <cit.>.]
Two forcings
are equivalent if and only if they have isomorphic Boolean completions.
Suppose $\iota:\PP\to\QQ$ is a dense embedding between forcings $\PP$ and $\QQ$.
Given a $\PP$-generic filter $G$ over $V$,
the upwards closure $H$ of $\iota(G)$ in $\QQ$ is a $\QQ$-generic filter over $V$, and $G=\iota^{-1}(H)$.
$(\sigma^\iota)^G=\sigma^H$ for each $\QQ$-name $\sigma$.
Conversely, suppose $H$ is a $\QQ$-generic filter over $V$. Then $G=\iota^{-1}(H)$ is a $\PP$-generic filter over $V$, and $H$ is equal to the upwards closure of $G$ in $\QQ$. In both of the above cases, we have $V[G]=V[H]$.

The following definition of the forcing for adding Cohen subsets is non-standard but essential in several arguments below.It is essential throughout Section <ref> that we use this non-standard definition

Suppose that $\kappa$ is a regular uncountable cardinal.
\beginenumerate-(a)

* $\Add(\kappa,1)$ is defined as the forcing

=\{p:\alpha\to\kappa\,\mid\, \alpha<\kappa\},$$

ordered by reverse inclusion.

* $\Add(\kappa,\xi)$ is defined as the ${<}\kappa$-support product $\prod_{i<\xi}\Add(\kappa,1)$ for any ordinal $\xi$.
\endenumerate-(a)

We use the following convention.
For a given $y\in{}^{\kappa}\kappa$,
we sometimes confuse $y$ with the set
$\kkppred y=\{t\in{}^{<\kappa}\kappa: t\subsetneq y\}$.

For example, we say that $y$ is $\Add(\kappa,1)$-generic if and only if $\kkppred y$ is an $\Add(\kappa,1)$-generic filter.

Moreover, if $y$ is $\Add(\kappa,1)$-generic and $\sigma$ is an $\Add(\kappa,1)$-name, let $\sigma^y:=\sigma^{\kkppred y}$.

Also let $\prod_{i<\xi}y_i$ denote $\prod_{i<\xi}\kkppred{y_i}$ whenever $\langle y_i:i<\xi\rangle$ is a sequence of elements of ${}^{\kappa}\kappa$.

The next two lemmas contain
standard facts about adding Cohen subsets and collapse forcings.

will only need this for uncountable $\kappa$
Suppose that $\kappa$ is an uncountable cardinal
inserted pagebreak
such that $\kappa^{<\kappa}=\kappa$.
If $\PP$ is a non-atomic ${<}\kappa$-closed forcing of size $\kappa$, then $\PP$ has a dense subset which is isomorphic to the dense subforcing

$$\Addd(\kappa,1):=\{p\in \Add(\kappa,1):\: \dom(p)\in \Succ\}$$

of $\Add(\kappa,1)$.
In particular, $\PP$ is equivalent to $\Add(\kappa,1)$.

We omit the proof of Lemma <ref>, since is straightforward and well known.

Let $\kappa$ be a regular cardinal.
Suppose that $\nu>\kappa$ is a cardinal with $\nu^{<\kappa}=\nu$. Let $\PP$ be a separative ${<}\kappa$-closed forcing of size $\nu$ which forces that $\nu$ has size $\kappa$.
Then $\PP$ has a dense subset which is isomorphic to the dense subforcing

$$\mathrm{Col}^*(\kappa,\nu):=\{p\in \Col(\kappa,\nu):\: \dom(p)\in\Succ\}$$

of $\Col(\kappa,\nu)$.
In particular, $\PP$ is equivalent to $\Col(\kappa,\nu)$.

Lemma <ref> can be proven
using an adaptation of the proof of
which can be found in the proof of <cit.>.

We will use the following notation
for subforcings of the Lévy collapse $\Col(\kappa,\lle\lambda)$.
Suppose that $\kappa<\lambda$ are cardinals, $\alpha\leq\lambda$ and $I\subseteq\lambda$ is not an ordinal (to avoid a conflict with the notation for the standard collapse).
Then let $\PP_\alpha:=\Col(\kappa,\lle\alpha)$

\[
\PP_I:=\Col(\kappa,I):=\{p\in \Col(\kappa,\lle\lambda):\: \dom{p}\subseteq I\times \kappa\}.
\]

The notation $\PP_I=\Col(\kappa,I)$ will be used for intervals $I$, for which we use the standard notation

\begin{align*}
(\alpha,\gamma)&:=\{\beta\in \Ord: \alpha<\beta<\gamma\}
\\
[\alpha,\gamma)&:=\{\beta\in \Ord: \alpha\leq\beta<\gamma\}.
\end{align*}

Let $\PP^\alpha:=\PP_{[\alpha,\lambda)}$

for all $\alpha<\lambda$.
If $G$ is a $\PP_\lambda$-generic filter over $V$, write
$G_\alpha:=G\cap\PP_\alpha$, $G_I:=G\cap\PP_I$
and $G^\alpha:=G\cap \PP^\alpha$.

The next corollary follows easily from Lemma <ref>
using a straightforward analogue of the proof of
<cit.>.[See also <cit.>.
asserts that the conclusion of Corollary <ref> holds when $\gamma=0$, under the weaker assumption that
$\lambda>\kappa$ is a cardinal such that $\mu^{<\kappa}<\lambda$ for all cardinals $\mu<\lambda$.]

Let $\lambda>\kappa$ be an inaccessible cardinal and $\gamma<\lambda$.
If $\PP$ is a $\lle\kappa$-closed forcing of size $\lle\lambda$,
then the forcings $\PP\times\PP_\lambda$ and
are equivalent.

Suppose that $\PP$, $\QQ$ are forcings.
\beginenumerate-(a)

A complete embedding $i: \PP\to \QQ$ is a homomorphism with respect to $\leq,\,\perp$ and $\one$ with the property that for every $q\in\QQ$, there is a condition $p\in \PP$ (called a reduction of $q$ to $\PP$) such that for every $r\leq p$ in $\PP$, $i(r)$ is compatible with $q$.

$\PP$ is a complete subforcing of $\QQ$
if $\PP$ is a subforcing of $\QQ$ and the inclusion map $\id_{\PP}:\PP\to\QQ$ is a complete embedding.

Suppose that $i: \PP\to \QQ$ is a complete embedding and $G$ is $\PP$-generic over $V$.
The quotient forcing $\QQ/G$ for $G$ in $\QQ$ is defined as the subforcing

\[\QQ/G:=\{q\in \QQ:\: \forall p\in G\ \,i(p)\compat q\}\]

of $\QQ$.
Moreover, we fix a $\PP$-name $\QQ/\PP$ for  for the quotient forcing for $\dot G$ in $\PP$, where $\dot{G}$ is the canonical $\PP$-name for the $\PP$-generic filter. We also refer to $\QQ/\PP$ as (a name for) the quotient forcing for $\PP$ in $\QQ$.
\endenumerate-(a)

We recall the following standard facts.[See e.g. <cit.> and
Exercises (C7), (C8), (D4) and (D5) in

Suppose $\PP$ and $\QQ$ are forcings and $i:\PP\to\QQ$.
\beginenumerate-(1)

If $i$ is a homomorphism with respect to $\leq,\,\perp$ and $\one$, then
$i$ is a complete embedding if and only if
for all maximal antichains $A$ of $\PP$,
$i(A)$ is a maximal antichain of $\QQ$.
If $\PP$ and $\QQ$ are complete Boolean algebras, then $i$ is a complete embedding (in the sense of Definition <ref>) if and only if $i$ is
an injective complete homomorphism of Boolean algebras

[A homomorphism between Boolean algebras is called complete if it preserves arbitrary infima and suprema.]

Any complete embedding $i:\PP\to\QQ$ defines a complete embedding
If $\PP$ and $\QQ$ are separative, we may assume that $i\subseteq j$.
In particular, if $\PP$ is a complete subforcing of $\QQ$ and $\QQ$ is separative,
[Note that complete subforcings of separative forcings are also separative.]
we may assume that $\BB(\PP)\subseteq\BB(\QQ)$.
Then $\BB(\PP)$ is a complete Boolean subalgebra of $\BB(\QQ)$ by <ref>.
If $\QQ$ is separative, then any complete subforcing $\PP$ of $\QQ$ is also separative. Proof: Let $p,q\in \PP$ with $p\not\leq q$. Since $\QQ$ is separative, take $s\in \QQ$ with $s\leq p$ and $s\perp q$. Let $r$ be a reduction of $s$ to $\PP$. Then $r\compat_{\PP} p$ since $\id$ is $\perp$-preserving and $r\perp q$ since $r$ is a reduction of $s$ (and $q\in\PP$).

Suppose $i$ is a complete embedding.
Let $G$ be a $\PP$-generic filter over $V$.
If $D$ is a dense subset of $\QQ$, then $D\cap\QQ/G$ is a dense subset of $\QQ/G$.
all $\QQ/G$-generic filters $H$ over $V[G]$ are also $\QQ$-generic filters over $V$, and $G=i^{-1}(H)$ for all such $H$.
if $H$ is $\QQ$-generic over $V$ and $G=i^{-1}(H)$, then $G$ is a $\PP$-generic filter over $V$ and $H$ is $\QQ/G$-generic over $V[G]$.
Furthermore, under either of the above assumptions, we have
\endenumerate-(1)

(Philipp:) I think both `natural projection' and 'retraction' are nice.
Given complete Boolean algebras $\PP$ and $\QQ$ and a complete embedding $i:\PP\to\QQ$,
the retraction associated to $i$ is the map
defined by letting

\[
\pi_i(q):=\bigwedgepo\PP\{p\in\PP: i(p)\geq q\}
\]

for all  $q\in \QQ$.

The following lemma is standard, but we include a proof for the reader.

Suppose $i:\PP\to\QQ$ is a complete embedding between the complete Boolean algebras $\PP$ and $\QQ$.
\beginenumerate-(1)

*  For all $q\in\QQ$, $\pi_i(q)$ is the largest reduction of $q$ to $\PP$.

If $G$ is a $\PP$-generic filter over $V$, then $q\in\QQ/G$ if and only if $\pi_i(q)\in G$.
\endenumerate-(1)

Let $q\in\QQ$.
First, observe that $i(p)\perp q$ if and only if $p\perp \pi_i(q)$ holds for all $p\in\PP$.
Indeed, given $p\in\PP$, we have
$i(p)\perp q$
$\Longleftrightarrow$ $i(p)\wedge q=\zero$
$\Longleftrightarrow$ $\neg i(p)\geq q$.
By the definition of $\pi_i$ and since $i$ is a Boolean homomorphism, the last statement is equivalent to
$\neg p\geq \pi_i(q)$,
and is therefore also equivalent to
$p\wedge \pi_i(q)=\zero$ and to

The previous observation easily implies that $\pi_i(q)$ is a reduction of $q$ to $\PP$.
To see that $\pi_i(q)$ is the largest reduction of $q$ to $\PP$, suppose that
$p\in\PP$ and $p\not\leq\pi_i(q)$.
Let $r:=p\wedge\neg\pi_i(q)$.
Then $p\geq r\geq\zero_\PP$ and $r\perp \pi_i(q)$, and therefore $i(r)\perp q$.
Thus, $r$ witnesses that
$p$ is not a reduction of $q$ to $\PP$.

Moreover, if $G$ is a $\PP$-generic filter over $V$ and $q\in\QQ$,
$p\perp \pi_i(q)$  for all $p\in\PP$
$\pi_i(q)\in G$.

As usual, by a name $\sigma$ for an element of ${}^\kappa\kappa$, we mean a name $\sigma$ such that $\one\forces\sigma\in {}^\kappa\kappa$.
By a name $\sigma$ for a new object, we mean a name $\sigma$ such that $\one\forces\sigma\notin \check V$.
In general, if $\varphi(v)$ is any formula with one variable $v$, then by a name $\sigma$ for an object with property $\varphi$, we mean a name $\sigma$ such that $\one\forces \varphi(\sigma)$.

We will use the following notation when working with quotient forcings induced by names.

Suppose $\QQ$ is a complete Boolean algebra and $\sigma$ is a $\QQ$-name for a subset of a set $x$.
Let $\BB(\sigma):=\BB^{\QQ}(\sigma)$ denote the complete Boolean subalgebra of $\QQ$ that is completely generated by
the set of Boolean values $\{\boolvalpo{y\in \sigma}{\QQ}: y\in x\}$.

The following lemma is standard, but we include a proof for the reader.

Suppose that $\QQ$ is a complete Boolean algebra and $\sigma$ is a $\QQ$-name for a subset of a set $x$.
If $H$ is a $\QQ$-generic filter over $V$, then
$V\big[\sigma^H\big]=V\big[\BB^{\QQ}(\sigma)\cap H\big]$.

Let $A:=\{\boolval{y\in\sigma}_{\QQ}:y\in x\}$.
Then $V\big[\BB^{\QQ}(\sigma)\cap H\big]=V\big[A\cap H\big]$
by <cit.>.
$V\big[A\cap H\big]=V\big[\sigma^H\big]$
$\Longleftrightarrow$ $\boolvalpo{y\in\sigma}\QQ\in H$
$\Longleftrightarrow$ $\boolvalpo{y\in\sigma}\QQ\in A\cap H$
for all $y\in x$.

In the next definition and lemmas, suppose that $\QQ\in V$ is a forcing, $q\in\QQ$ and $\sigma\in V$ is a $\QQ$-name for an element of ${}^\kappa\kappa$.

\beginenumerate-(a)

* $\sigma_{[q]}:=\bigcup\{t\in{}^{<\kappa}\kappa: q\forces t\subseteq\sigma\}$.

* $\sigma_{(q)}:=\{(\alpha,\beta):q\forces(\alpha,\beta)\in\sigma\}$.

* The tree $T^{\sigma, q}$ of possible values for $\sigma$ below $q$ is defined as follows:
Equivalently, $t\in T^{\sigma, q}$ $\Leftrightarrow$ $r\forces t\subseteq \sigma$ for some $r\leq q$

\[
T^{\sigma, q}:=
\{t\in{}^{<\kappa}\kappa\::\: \exists r\leq q\ \ t\subseteq \sigma_{[r]}\}.\]

\endenumerate-(a)
For clarity, we at times write $\sigma_{[q,\QQ]},\ \sigma_{(q,\QQ)}$ and $T^{\sigma,q}_{\QQ}$ instead of $\sigma_{[q]},\ \sigma_{(q)}$ and $T^{\sigma,q}$, respectively.

The next two lemmas list basic properties of $\sigma_{[q]}$ and $T^{\sigma,q}$.
The first one follows immediately from the definitions.

\beginenumerate-(1)

* $\sigma_{[q]}\subseteq \sigma_{(q)}$, and
$\sigma_{[q]}=\sigma_{(q)}$ if and only if $\dom(\sigma_{(q)})$ is an ordinal.

* $\sigma_{[q]}\restr\alpha=\sigma_{(q)}\restr\alpha$ for all ordinals $\alpha$ with $\alpha\subseteq\dom(\sigma_{(q)})$.
\endenumerate-(1)

\beginenumerate-(1)

If $q\not\forces_{\QQ} \sigma \in\check{V}$,
inserted pagebreak
then $\sigma_{[q]}\in{}^{<\kappa}\kappa$ equals the stem of  $T^{\sigma,q}$.

$q\forces^V_{\QQ} \sigma\in[T^{\sigma,q}].$

If $S$ is a subtree of ${}^{<\kappa}\kappa$ with $q\forces^V_{\QQ} \sigma\in[S]$, then $T^{\sigma,q}\subseteq S$.

If $M\supseteq V$ is a transitive model of $\ZFC$ with
$({}^{<\kappa}\kappa)^V=({}^{<\kappa}\kappa)^M$, then
${(T^{\sigma, q})}^V={(T^{\sigma, q})}^M.$
If $\QQ$ is a $\lle\kappa$-closed forcing
and $q\forces\sigma\notin\check{V}$,
then $T^{\sigma,q}$ is a $\kappa$-perfect tree.

PROOF: $T:=T^{\sigma,q}$ is cofinally splitting because $\sigma$ is a name for a new element of ${}^\kappa\kappa$.
To obtain a $\kappa$-perfect (in the stronger sense) subtree containing a given node $s\in T$: use that $T$ is cofinally splitting at successor stages of defining the tree, and use the fact that
$\QQ$ is $\lle\kappa$-closed at limit stages.
It's probably enough to assume that $\QQ$ is $\lle\kappa$-strategically closed.
\endenumerate-(1)

<ref>–<ref> are immediate.
<ref> holds since the formula “$q\forces t\subseteq \sigma$” is absolute between $M$ and $V$, as it  can be defined by a recursion which uses only absolute concepts <cit.>.

§ DIHYPERGRAPHS AND HOMOMORPHISMS

We discuss basic facts about colorings, continuous homomorphisms and
the open dihypergraph dichotomy in Subsection <ref>.
We characterize $H$-independence for box-open dihypergraphs $H$ at the level of subtrees of ${}^{<\kappa}\kappa$ in Section <ref>.

We provide a characterization of
the existence of continuous homomorphisms from $\dhH\ddim$ to $H$
via certain strict order preserving maps in Subsection <ref>.
These facts will be used throughout the paper.
The relation
of fullness between dihypergraphs that is introduced in Subsection <ref>
compares the induced
$\kappa$-ideals of $\kappa$-colorable sets and, as we show, allows us
to compare the existence of continuous homomorphisms
for relatively box-open dihypergraphs, too.
[See Lemma <ref>.]
In particular,
equivalent relatively box-open dihypergraphs give rise to
the same instances of the open dihypergraph dichotomy.

§.§ Colorings and homomorphisms

Throughout this subsection,
we assume that $\ddim\geq 2$ is an ordinal.

We first observe
that the two options in the definition of $\ODD\kappa H$ are mutually exclusive.

Let $H$ be a $\ddim$-dihypergraph on a topological space $X$.
Suppose that there is a homomorphism $f:{}^\kappa\ddim\to X$ from $\dhH\ddim$ to $H$.
\beginenumerate-(1)

*  $H$ does not have a $\kappa$-coloring.

$H\restr f(N_t)$ does not have a $\kappa$-coloring for any $t\in{}^{<\kappa}\ddim$.
Stating <ref> explicitly will be useful in some of the arguments below.
\endenumerate-(1)

The proof of <ref> is a straightforward analogue of the proof for the $\kappa=\omega$ case
in <cit.>*Theorem 1.
We give the details for completeness.
Suppose $c':X\to\kappa$ is a $\kappa$-coloring of $H$, and let $c:=c'\comp f$. Since $f$ is a homomorphism from $\dhH\ddim$ to $H$, $c$ is a $\kappa$-coloring of $\dhH\ddim$.
Construct by recursion a continuous increasing sequence
$\langle t_\alpha:\alpha<\kappa\rangle$ such that
$t_\alpha\in{}^{\alpha}\ddim$ and
$c^{-1}(\{\alpha\})\cap N_{t_{\alpha+1}}=\emptyset$
for each $\alpha<\kappa$.
Use the
$\dhH\ddim$-independence of $c^{-1}(\{\alpha\})$ at
stage $\alpha+1$ of the construction.
$\bigcup_{\alpha<\kappa} t_\alpha$ is an element of ${}^\kappa\ddim$
which is not in $c^{-1}(\{\alpha\})$ for any $\alpha<\ddim$,
a contradiction.

<ref> follows from <ref>, since
the map
$f_t:{}^\kappa\ddim\to f(N_t);\; x\mapsto f(t\conc x)$
is a homomorphism from $\dhH\ddim$ to $H\restr f(N_t)$
for any given $t\in{}^\kappa\ddim$.

Note that
<ref> implies $|X|>\kappa$ and <ref> implies
$\left|f (N_t)\right|>\kappa$
for all $t\in{}^{<\kappa}\ddim$.
We did not need to assume that $f$ is continuous or that $H$ is box-open.
Thus, if $\ODD\kappa H$ holds and there exists an arbitrary homomorphism from $\dhH \ddim$ to $H$, then there also exists a continuous one.
This is analogous to the fact that for any subset $X$ of ${}^\kappa\kappa$ that satisfies the $\kappa$-perfect set property, the existence of an injective function from ${}^\kappa 2$ to $X$ already implies the existence of a continuous injective function.

Suppose $\ddim<\kappa$ and $X\subseteq{}^\kappa\kappa$.
\beginenumerate-(1)

If $U$ is a box-open subset of ${}^\ddim ({}^\kappa\kappa)$
then $U\in\defsets\kappa$.

$\ODD\kappa\ddim(X)$ is equivalent to $\ODD\kappa\ddim(X,\defsets\kappa)$.
\endenumerate-(1)

For <ref>, note that $U$ is given by a sequence $x: \kappa\to {}^\ddim({}^{<\kappa}\kappa)$ since the base of the topology has size $|{}^\ddim({}^{<\kappa}\kappa)|=\kappa$. Since $x$ can be coded by an element of ${}^\kappa\kappa$, we have $U\in \defsets\kappa$.
<ref> follows from <ref>.

The next two lemmas give reformulations of $\ODD\kappa\ddim(\defsetsk)$ and $\ODD\kappa\ddim(\defsetsk,\defsetsk)$.
Recall that $H$ is a relatively box-open dihypergraph if it is box-open on its domain $\domh H$.

$\ODD\kappa\ddim(\defsetsk)$ is equivalent to the statement that $\ODD\kappa H$ holds for all
relatively box-open $\ddim$-dihypergraphs
$H$  on ${}^\kappa\kappa$
with $\domh H\in\defsetsk$.

This follows from the observation that a dihypergraph
$H$ is relatively box-open with $\domh H\in\defsetsk$
if and only if
$H=H'\restr X$ for some
subset $X\in\defsetsk$ of ${}^\kappa\kappa$ and some
box-open $\ddim$-dihypergraph $H'$ on ${}^\kappa\kappa$.
To see the direction from right to left, note that $\domh{H'\restr X}$ is a relatively open subset of $X$,
and hence it is in $\defsetsk$ if $X$ is in $\defsetsk$.

Suppose that $d\leq\kappa$. Then
$\ODD\kappa\ddim(\defsetsk,\defsetsk)$ is equivalent to the statement that $\ODD\kappa H$ holds for all
relatively box-open $\ddim$-dihypergraphs
$H\in \defsetsk$ on ${}^\kappa\kappa$.

This follows from the equivalence of the following statements for any $\ddim\leq\kappa$ and any $\ddim$-dihypergraph $H$ on ${}^\kappa\kappa$:
\beginenumerate-(a)

$H\in \defsetsk$ is relatively box-open.

$H=H'\restr X$ for some
subset $X\in\defsetsk$ of ${}^\kappa\kappa$ and some
box-open $\ddim$-dihypergraph $H'\in\defsetsk$ on ${}^\kappa\kappa$.
\endenumerate-(a)
For <ref> $\Rightarrow$ <ref>,
note that
$X:=\domh H\in \defsetsk$ and

$$H':=\bigcup\{\prod_{\alpha<\ddim}N_{t_\alpha}:\, \prod_{\alpha<\ddim}(N_{t_\alpha}\cap X)\subseteq H\}\cap\dhK\ddim{{}^\kappa\kappa} \in\defsetsk$$

is a box-open $\ddim$-dihypergraph on ${}^\kappa\kappa$
with $H=H'\restr X$.
The converse is clear.

Let $f:X\to Y$ be a continuous surjection between subsets $X$ and $Y$ of ${}^\kappa\kappa$.
\beginenumerate-(1)

$\ODD\kappa\ddim(X)$ implies $\ODD\kappa\ddim(Y)$.

If $f$ can be extended to a continuous function “continuous” is needed  in the proof, and it implies $g\in\defsetsk$
$g: {}^\kappa\kappa\to {}^\kappa\kappa$,

then $\ODD\kappa\ddim(X,\defsets\kappa)$ implies $\ODD\kappa\ddim(Y,\defsets\kappa)$.
\endenumerate-(1)

$\ODD\kappa\ddim(X)$ is equivalent to the statement that $\ODD\kappa I$ holds for all box-open $\ddim$-dihypergraphs $I$ on $X$.
For a given box-open $\ddim$-dihypergraph $H$ on $Y$, let

\[H_f={{(f^\ddim)}^{-1}(H)}
=\{\langle x_\alpha:\alpha<\ddim\rangle\in{}^\ddim X:
\langle f(x_\alpha):\alpha<\ddim\rangle\in H\}.
\]

Since ${f^\ddim}$ is a continuous map between the spaces ${}^\ddim X$ and ${}^\ddim Y$,
where both spaces are
equipped with the box-topology,
$H_f$ is a box-open $\ddim$-dihypergraph on $X$.

$\ODD\kappa {H_f}$ implies $\ODD\kappa H$.

Note that $f$ is a continuous homomorphism from $H_f$ to $H$.
Thus, if $h:{}^\kappa\ddim\to X$ is a continuous homomorphism from $\dhH\ddim$ to $H_f$, then $h\comp f$ is a continuous homomorphism from $\dhH\ddim$ to $H$.

On the other hand, if $c:X\to\kappa$ is a $\kappa$-coloring of $H_f$, then the map
$c':Y\to\kappa$ defined by letting
\min\big\{\alpha:\:f^{-1}(\{y\})\cap {(c)}^{-1}(\{\alpha\})\neq\emptyset\big\}$
for each $y\in Y$
is a $\kappa$-coloring of $H$.

<ref> follows immediately from the previous claim.
For <ref>, suppose that $H'\in \defsets\kappa$ is a box-open $\ddim$-dihypergraph on ${}^\kappa\kappa$.
Let $H:=H'{\restr}Y$.
It suffices to show
$\ODD\kappa H$.
Let $H_f:={(f^\ddim)}^{-1}(H)$ and $H'_g={(g^\ddim)}^{-1}(H')$.
Since $g$ is continuous,
is a box-open needs $g$ to be continuous
$\ddim$-dihypergraph on ${}^\kappa\kappa$.
Since we assume
$\ODD\kappa\ddim(X,\defsetsk)$, we have $\ODD\kappa\ddim(X,H'_g)\Longleftrightarrow\ODD\kappa{H'_g{\restr}X}\Longleftrightarrow\ODD\kappa{H_f}$.
Thus $\ODD\kappa H$ by the previous claim.
Short explicit proof of $H'_g\restr X=H_f$ (I think it can be omitted though): $\bar x\in H'_g\restr X$ if and only if ($\bar x\in{}^\ddim X$ and $H'\ni g(\bar x)=f(\bar x)$). In the last equality, we use that $f=g\restr X$. Since $H=H'\restr Y$ and $\ran(f)\subseteq H$, the latter condition holds if and only if ($\bar x\in{}^\ddim X$ and $f(\bar x)\in H$), i.e., if and only if $\bar x\in H_f$

We next show that $\ODD\kappa\ddim(X)$ implies $\ODD\kappa c(X)$
for all $c\subseteq d$
with $ |c|\geq 2$.
We will need the following lemma.
Let $\proj_{\ddim,c}$ denote
the projection of $\ddim$-sequences onto
coordinates in $c$,
for any sequence
$\bar x=\langle x_i:i\in\ddim\rangle$,

\[\proj_{\ddim,c}(\bar x)=\bar x\restr c=\langle x_i: i\in c\rangle.\]

Recall that $\dhK\ddim X$ is the complete $\ddim$-hypergraph on $X$ and
$\dhII\ddim X$ consists of
the injective elements of $\dhK\ddim X$.

Suppose that $c\subseteq d$ and $|c|\geq 2$.
Let $H$ be a

on a topological space $X$,
and let
$I:=\proj_{\ddim, c}^{-1}(H){\restr}X$.
Equivalently, $I:=\proj_{\ddim, c}^{-1}(H)\cap\dhK\ddim X$, since
$\proj_{\ddim, c}^{-1}(\dhK c X)\cap{}^\ddim X\subseteq\dhK\ddim X$.
\beginenumerate-(i)

$I$ is a $\ddim$-dihypergraph on $X$.

$H$ is box-open on $X$ if and only if $I$ is box-open on $X$.
In fact, if $c$ is finite and $H$ is open on $X$,
then $I$ is product-open on $X$.
[I.e., $H$ is an open subset of ${}^\ddim X$ in the product topology.]
This is not true for closed dihypergraphs, since maybe $I\subsetneq \proj_{\ddim, c}^{-1}(H\cup\dhC c X)\cap\dhK\ddim X$.

If $c,X\in\defsetsk$, then $H\in\defsetsk$ if and only if $I\in\defsetsk$.

Suppose that $Y\subseteq X$.
\beginenumerate-(a)

* $\dhK\ddim Y\not\subseteq I$.

* If $|Y|\geq|\ddim|$, then
$\dhII c Y\subseteq H$ if and only if $\dhII \ddim Y\subseteq I$.
\endenumerate-(a)

Suppose that $Y\subseteq X$.
\beginenumerate-(a)

* $Y$ is $H$-independent if and only if $Y$ is $I$-independent.

* If $|Y|\geq|\ddim|$, then
$\dhII c Y\cap H=\emptyset$ if and only if $\dhII\ddim Y\cap I=\emptyset$.
\endenumerate-(a)

A function $f:X\to \kappa$ is a $\kappa$-coloring of $H$ if and only if it is a $\kappa$-coloring of $I$.

There exists a continuous homomorphism from $\dhH c$ to $H$ if and only if there exists a continuous homomorphism from $\dhH\ddim$ to $I$.

$\ODD\kappa{H}$ $\Longleftrightarrow$ $\ODD\kappa{I}$.
\endenumerate-(i)

are easy to show.
<ref> follows from <ref>.

To show <ref>,
suppose first that $g:{}^\kappa\ddim\to X$ is a continuous homomorphism from $\dhH\ddim$ to $I$.
Let $f:=g\restr{}^\kappa c$.
It is easy to see that $f$ is continuous.
To see that $f$ is a homomorphism from $\dhH c$ to $H$,
suppose that $\bar y\in \dhH c$.
Then there exists $\bar x\in{}\dhH\ddim$ such that $\proj_{\ddim, c}(\bar x)=\bar y$.
Since $g$ is a homomorphism from $\dhH\ddim$ to $I$,
we have $g^\ddim(\bar x)\in I$.
$f^c(\bar y)=
g^c(\bar y)=
\proj_{\ddim,c}\big(g^\ddim(\bar x)\big)
\in H,$
as required.
Conversely, suppose that $f:{}^\kappa c\to X$ is a continuous homomorphism from $\dhH c$ to $H$.
Fix a map $k: d\to c$ with $k{\upharpoonright}c = \id_c$ and define $g:{}^\kappa\ddim\to X$ by letting $g(x):=f(k\circ x)$.

It is easy to see that $g$ is continuous.
To see that $g$ is a homomorphism from $\dhH \ddim$ to $I$,
suppose that $\bar y= \langle y_i : i\in \ddim\rangle \in \dhH \ddim$.
Then $g^\ddim(\bar{y})=\langle f(k\circ y_i) : i\in \ddim\rangle$ and its projection equals $\langle f(k\circ y_i) : i\in c\rangle$.
Suppose that $y_i$ extends $t^\smallfrown \langle i\rangle$ for all $i\in d$.
Then $k\circ y_i$ extends $(k\circ t)^\smallfrown \langle i\rangle$ for all $i\in c$ and hence $\langle f(k\circ y_i) : i\in c\rangle\in H$.
Thus $g^\ddim(\bar{y})\in I$.
<ref> follows from <ref> and <ref>.

ODD^kappa implies definable ODD^ddim
Suppose that $c\subseteq d$, $|c|\geq 2$
and $X$ is a subset of ${}^\kappa\kappa$.
\beginenumerate-(1)

$\ODD\kappa\ddim(X)$ implies $\ODD\kappa c(X)$.

$\ODD\kappa\ddim(X,\defsetsk)$ implies $\ODD\kappa c(X,\defsetsk)$
if $c\in\defsetsk$.
\endenumerate-(1)

For <ref>, suppose $H$ is a box-open $c$-dihypergraph on ${}^\kappa\kappa$.
It suffices to show $\ODD\kappa {H{\restr}X}$.
$I=\proj_{\ddim, c}^{-1}(H){\restr}{}^\kappa\kappa$ is a box-open $d$-dihypergraph on ${}^\kappa\kappa$ by <ref> and <ref> in the previous lemma.
By $\ODD\kappa\ddim(X)$ we have  $\ODD\kappa {I{\restr}X}$ and then obtain $\ODD\kappa {H{\restr}X}$ from <ref>.
The proof of <ref> is similar, but also uses <ref>.

In particular, $\ODD\kappa\kappa(X,\defsetsk)$ implies $\ODD\kappa c(X)$ for all $2\leq c<\kappa$ and $X\subseteq{}^\kappa\kappa$ by
Lemmas <ref> and <ref>.

§.§ Independent trees

We assume throughout this subsection that $d$ is a set
of size at least $2$.
We characterize
independence for box-open dihypergraphs
at the level of subtrees of ${}^{<\kappa}\kappa$.

Suppose $X$ is a subset of ${}^\kappa\kappa$ and $H$ is a $\ddim$-dihypergraph on ${}^\kappa\kappa$.
A subtree $T$ of ${}^{<\kappa}\kappa$
is an $(X,H)$-independent tree if
$\prod_{i\in\ddim}(N_{t_i}\cap X)\not\subseteq H$
for all sequences $\langle t_i:i\in\ddim\rangle \in{}^\ddim T$.

We omit $X$ if $X={}^\kappa\kappa$, i.e., $T$ is called $H$-independent if it is $({}^\kappa\kappa,H)$-independent.

We will most often work with $H$-independent trees.
Note that if $T$ is an $(X,H)$-independent tree, then $T\subseteq T(X)$ because otherwise, there exists some $t\in T$ with $N_t\cap X=\emptyset$.

Suppose $H$ is a $\ddim$-dihypergraph on a subset $X$ of ${}^\kappa\kappa$.
\beginenumerate-(1)

If $H$ is box-open on $X$ and $T$ is an $(X,H)$-independent tree, then $[T]$ is an $H$-independent set.

If $Y\subseteq X$ is an $H$-independent set, then $T(Y)$ is an $(X,H)$-independent tree.
\endenumerate-(1)

For <ref>, suppose that $[T]$ is not $H$-independent, and
let $\langle y_i:i\in\ddim\rangle \in{}^\ddim[T]\cap H$.
Since $H$ is box-open on $X$,
there exist nodes $t_i\subsetneq y_i$
such that
$\prod_{i\in\ddim}(N_{t_i}\cap X)\subseteq H$.
Then $t_i\in T$ for all $i\in\ddim$, so $T$ is not $(X,H)$-independent.

For <ref>, suppose that $Y\subseteq X$ and $T(Y)$ is not $(X,H)$-independent.
Take $\langle t_i:i\in\ddim\rangle$ in ${}^\ddim T(Y)$ such that
$\prod_{i\in\ddim}(N_{t_i}\cap X)\subseteq H$.
For all $i\in\ddim$,
fix $y_i\in Y$ with $t_i\subseteq y_i$.
Then $\langle y_i:i\in\ddim\rangle$ is a hyperedge of $H$.

Suppose $H$ is a box-open $\ddim$-dihypergraph on a subset $X$ of ${}^\kappa\kappa$.
If $Y\subseteq X$ is $H$-independent, then $\closure Y$
is $H$-independent.
[Recall that $\closure Y$ denotes the closure of $Y$ in ${}^\kappa\kappa$.]

The statement follows from the previous lemma, since $\closure Y=[T(Y)]$.

§.§ Order homomorphisms

We assume throughout this subsection that $d$ is a discrete topological space of size at least $2$.
We first characterize the existence of a continuous homomorphism from $\dhH\ddim$ to
box-open dihypergraphs
using the following types of strict order preserving maps.
We use terminology from Subsection <ref>.

Suppose $X\subseteq{}^\kappa\kappa$ and $H$ is a $\ddim$-dihypergraph on ${}^\kappa\kappa$.
A strict order preserving map $\iota:{}^{<\kappa}\ddim\to{}^{<\kappa}\kappa$
is an order homomorphism for $(X,H)$
\beginenumerate-(a)

$\ran([\iota])\subseteq X$, and

$\prod_{i\in \ddim}(N_{\iota(t\conc\langle i\rangle)}\cap X)\subseteq H$
for all $t\in{}^{<\kappa}\ddim$.
\endenumerate-(a)

We omit $X$ if it equals $\domh H$, i.e.,
$\iota$ is an order homomorphism for $H$ if it is
an order homomorphism for $(\domh H,H)$.

We will most often work with order homomorphisms for $H$.
If $\domh H\subseteq X$, then any order order homomorphism $\iota$ for $(X,H)$ is also an order homomorphism for $H$,
since <ref> and <ref> imply $\ran([\iota])\subseteq \domh H$.

If $\domh{H}\subseteq X$ is relatively open,
[For instance, if $H$ is box-open on $X$.]
then the converse holds:
if there exists an order homomorphism $\iota$ for $H$, then there exists one for $(X,H)$.
The special case when $H$ is box-open on $X$ also follows from Lemma <ref> below.
To see this, one constructs by recursion a strict order preserving map $e:{}^{<\kappa}\ddim\to{}^{<\kappa}\ddim$ such that $\theta:=\iota\circ e$ satisfies $N_{\theta(t\conc\langle i\rangle))}\cap X\subseteq
N_{\iota(e(t)\conc\langle i\rangle)}\cap \domh{H}$ for all $t\in{}^{<\kappa}\kappa$;
then $\theta$ is as required.

Suppose $X$ and $H$ are as in the previous definition
and $\iota:{}^{<\kappa}\ddim\to T(X)$.
We define the following $\ddim$-dihypergraphs on ${}^{<\kappa}\ddim$ and ${}^{<\kappa}\kappa$, respectively:
\beginenumerate-(1)

* $\HH^-_{{}^{<\kappa}\ddim}$ consists of all sequences in ${}^{<\kappa}\ddim$ of the form
$\langle t\conc\langle \alpha\rangle: \alpha<\ddim\rangle$.

* $H^-_X$ consists of all sequences $\langle u_\alpha:\alpha<\ddim\rangle$ in $T(X)$

with $\prod_{\alpha<\kappa} (N_{u_\alpha}\cap X)\subseteq H$.
[Note that a subtree $T$ of $T(X)$ is $(X,H)$-independent in the sense of Definition <ref> if and only if it is $H^-_X$-independent in the sense that $H^-_X\restr T=\emptyset$.]
\endenumerate-(1)
Using these definitions, <ref> in Definition <ref> is equivalent to the statement that
is a homomorphism from $\HH^-_{{}^{<\kappa}\ddim}$ to $H^-_X$.
This motivates the name “order homomorphism”.

Let $\HH_{\iota,X} :=
\bigcup_{t\in{}^{<\kappa}\ddim}\prod_{i\in \ddim} (N_{\iota(t\conc\langle i\rangle)}\cap X)
<ref> is further equivalent to the inclusion
$\HH_{\iota,X}\subseteq H$.
Note that $\HH_{\iota,X}$ is a box-open subset of ${}^\ddim X$.
It is a dihypergraph if and only if
$X\cap\bigcap_{i\in\ddim} N_{\iota(t\conc\langle i\rangle)}=\emptyset$ for all $t\in{}^{<\kappa}\ddim$ since otherwise it contains constant sequences.
For $\HH_{\iota,X}$ to be a dihypergraph, it suffices that for every $t\in {}^{<\kappa}\ddim$, there exist $i,j\in \ddim$ with $\iota(t\conc\langle i \rangle)\perp \iota(t\conc\langle j \rangle)$.
This condition is necessary if $\ran(\iota)\subseteq T(X)$,
$T(X)$ is ${\leq}|\ddim|$-closed if $|d|<\kappa$
and (b)
$X$ is closed and $T(X)$ is ${<}\kappa$-closed if $|d|=\kappa$.

The next lemma shows that for relatively box-open dihypergraphs $H$,
the existence of an order homomorphism for $H$
is equivalent to the existence of a continuous homomorphism from $\dhH\ddim$ to $H$.
We state a more general version for order homomorphisms for $(X,H)$ which implies
the previous statement in the special case $X=\domh H$.

Suppose $X\subseteq{}^\kappa\kappa$ and $H$ is a $\ddim$-dihypergraph on ${}^\kappa\kappa$.
\beginenumerate-(1)

If $\iota$ is an order homomorphism for $(X,H)$, then
$[\iota]$ is a continuous homomorphism from $\dhH\ddim$ to $H\restr X$.

If $H\restr X$ is box-open on $X$ and $f$ is
$X$ is important here, e.g when $\domh H\supseteq X$, an order homomorphism for $(X,H)$ is a stronger notion than an order homomorphism for $H$ whose range is a subset of $X$.
a continuous homomorphism
from $\dhH\ddim$ to $H\restr X$, then
there exists a continuous
See Definition <ref> <ref>.
order homomorphism $\iota$ for $(X,H)$
and a continuous strict $\wedge$-homomorphism
[See Definition <ref>. Note that $e:{}^{<\kappa}\ddim\to{}^{<\kappa}\kappa$  is a strict $\wedge$-homomorphism if and only if it is strict order preserving and $e(t\conc\langle i\rangle) \wedge e(t\conc\langle j\rangle) =e(t)$ for all $t\in{}^{<\kappa}\ddim$ and $i,\,j\in\ddim$ with $i\neq j$. It is easy to see that such a map $e$ is $\perp$-preserving, and hence $[e]$ is a homeomorphism onto its image by Lemma <ref> <ref>.]
with $[\iota]=f\comp [e]$.

[In general, it is not possible to find a strict order preserving map $\iota$ that induces the original function $f$ (i.e., $[\iota]=f$). For instance, let $f$ map $N_{\langle 0\rangle}$ onto $N_{\langle 0\rangle} \cup N_{\langle 1\rangle}$. Note that it is always possible to find an order preserving map $\iota$ that induces $f$ <cit.>*Lemma 3.7. But order preserving maps $\iota$ as in Definition <ref> <ref> for graphs $H$ are always $\perp$- and hence strict order preserving.]
\endenumerate-(1)
In particular, <ref> and <ref> apply to order homomorphisms for $H$ and continuous homomorphisms from $\dhH\ddim$ to $H$ by letting $X=\domh H$.

We first show <ref>.
Since $\iota$ is strict order preserving,
$[\iota]$ is a continuous function from ${}^{\kappa}\ddim$ to $X$
by Lemma <ref>.
It thus suffices to show that $[\iota]$ is a homomorphism from $\dhH\ddim$ to $H$.
Suppose $\bar x=\langle x_i:i\in\ddim\rangle\in\dhH\ddim$.
Let $u\in{}^{<\kappa}\ddim$ be such that
$u\conc\langle i\rangle\subseteq x_i$
for all $i\in\ddim$.
$\iota(u\conc\langle i\rangle)\subseteq [\iota](x_i)$
for all $i\in\ddim$, and therefore

\[
{[\iota]^\ddim}(\bar x)=
\langle[\iota](x_i):i\in\ddim\rangle\in
\prod_{i\in\ddim}(N_{\iota(u\conc\langle i\rangle)}\cap X)\subseteq H.
\]

To prove <ref>, let $f$ be a continuous homomorphism from $\dhH\ddim$ to $H\restr X$.

We construct
continuous strict order preserving
such that for all $t\in{}^{<\kappa}\ddim$ and $i\in\ddim$,
\beginenumerate-(i)

$e(t)\conc\langle i\rangle\subseteq e(t\conc\langle i\rangle)$,

$f(N_{e(t)})\subseteq N_{\iota(t)}$, and

$\prod_{i\in\ddim}(N_{\iota(t\conc\langle i\rangle)}\cap X)\subseteq H$.
\endenumerate-(i)
We construct $\iota(t)$ and $e(t)$ by recursion on $\lh(t)$.
Let $\iota(\emptyset):=e(\emptyset)=\emptyset$.
Let $t\in{}^{<\kappa}\ddim\,\setminus\{\emptyset\}$, and suppose that $\iota(u)$ and $e(u)$ have been defined for all $u\subsetneq t$.
If $\lh(t)\in\Lim$,
let $\iota(t):=\bigcup_{u\subsetneq t}\iota(u)$ and
$e(t):=\bigcup_{u\subsetneq t}e(u)$.
$N_{e(t)}=\bigcap_{u\subsetneq t}N_{e(u)}$
and <ref> holds for all $u\subsetneq t$,
we have

\[
\bigcap_{u\subsetneq t}f(N_{e(u)})\subseteq
\bigcap_{u\subsetneq t}N_{\iota(u)}
\]

Suppose $\lh(t)\in\Succ$, and let $u$ be the immediate predecessor of $t$.
We define
$\iota(u\conc\langle i\rangle)$
$e(u\conc\langle i\rangle)$
for each $i\in\ddim$ simultaneously as follows.
$x_i\in{}^{\kappa}\ddim$ extend $e(u)\conc\langle i\rangle$ for each $i\in\ddim$.
Since $f$ is a homomorphism from $\dhH\ddim$ to $H\restr X$, the sequence
$\langle f(x_i):i\in\ddim\rangle$ is a hyperedge of $H\restr X$.
Since $H\restr X$ is box-open on $X$, there exists a sequence
$\langle\iota(u\conc\langle i\rangle): i\in\ddim\rangle$
such that
$\iota(u)\subsetneq\iota(u\conc\langle i\rangle)\subsetneq f(x_i)$
and <ref> holds for $u$.
By the continuity of $f$, there exists for each $i\in\ddim$
an initial segment $e(u\conc\langle i\rangle)$ of $x_i$ such that $e(u)\conc\langle i\rangle\subseteq e(u\conc\langle i\rangle)$ and
<ref> holds for $u\conc\langle i\rangle$.

Suppose the continuous strict order preserving
maps $\iota$ and $e$ have been constructed.
$e$ is a strict $\wedge$-homomorphism by <ref>.
<ref> implies that
for all $x\in{}^\kappa\ddim$,

\[
\bigcap_{t\subsetneq x}f(N_{e(t)})\subseteq
\bigcap_{t\subsetneq x}N_{\iota(t)}
\]

Thus $[\iota]=f\comp[e]$, and therefore
$\ran([\iota])\subseteq\ran(f)\subseteq X$.
This fact and <ref> imply that $\iota$ is an order homomorphism for $(X,H)$.

Fix any club $C$ in $\kappa$.
The construction in the proof of Lemma <ref> <ref> can be easily modified to obtain $\lh(\iota(t))\in C$ for all $t\in {}^{<\kappa}\ddim$ in addition to the above properties.

If $H\restr X$ is box-open on $X$ and there is a continuous homomorphism from $\dhH\ddim$ to $H\restr X$, then
there exists an
order homomorphism $\iota$ for $(X, H)$
such that $\lh(\iota(t))$ is even for all $t\in{}^{<\kappa}\kappa$.

Since $ H\restr X$ is a box-open dihypergraph on $X$,
let $\psi$ be

order homomorphism for $(X, H)$
by Lemma <ref>.
We construct

strict order preserving maps
$e:{}^{<\kappa}\kappa\to{}^{<\kappa}\kappa$ so that
the following hold for all $t\in{}^{<\kappa}\kappa$ and $\alpha<\kappa$:
\beginenumerate-(i)

$\lh(\iota(t))$ is even,

\iota(t\conc\langle\alpha\rangle)$,

$\iota(t)\subseteq \psi(e(t))$.
\endenumerate-(i)
To construct $\iota$ and $e$, take
a map $s:{}^{<\kappa}\kappa\to T(\psi)$
such that
$\psi(t)\subseteq s(t)$
$\lh(s(t))$ is even
for all $t\in{}^{<\kappa}\kappa$.
We define $\iota(t)$ and $e(t)$ by recursion on $\lh(t)$ as follows.
Let $e(\emptyset):=\emptyset$ and $\iota(\emptyset):=s(\emptyset)$
If $\lh(t)\in\Lim$, let $e(t):=\bigcup_{u\subsetneq t} e(u)$ and
Now assume $e(t)$ and $\iota(t)$ have been constructed.
For each $\alpha<\kappa$,
let $\iota(t\conc\langle\alpha\rangle):=
and choose
so that

To see that $\iota$ has the required properties, note that $\lh(\iota(t))$ is always even and
$\iota$ is strict order preserving by the construction.
$\ran([\iota])\subseteq\ran([\psi])\subseteq X$ since
$[\iota]=[\psi\comp e]=[\psi]\comp[e]$ by <ref>.
$\iota$ is an order homomorphism for $(X, H)$ by <ref> and since $\psi$ is an order homomorphism.

§.§ Full dihypergraphs

In this subsection,
$\ddim$ and $c$ are discrete topological spaces of cardinality at least $2$.
We show that if $H$ and $I$ are relatively box-open
dihypergraphs on ${}^\kappa\kappa$,
$\ODD\kappa H\Longleftrightarrow\ODD\kappa I$
whenever $H\equivf I$ in the sense of the next definition.

Suppose that $H$ and $I$ are $\ddim$- and $c$- dihypergraphs, respectively, on

the same set $X$.

page break before sentence
\beginenumerate-(a)

$I$ is $H$-full ($H\leqf I$)
if $H\restr A$ admits a $\kappa$-coloring for every $I$-independent subset $A$ of $X$.
[Equivalently, $H\restr A$ admits a $\kappa$-coloring for all $A\subseteq X$ such that $I\restr A$ admits a $\kappa$-coloring.]

$H\equivf I$

if for all $A\subseteq X$,
$H\restr A$ admits a $\kappa$-coloring if and only if $I\restr A$ admits a $\kappa$-coloring.

\endenumerate-(a)

Note that $H\equivf I$ if and only if $H\leqf I$ and $I\leqf H$.

For example,
$H\leqf I$ if $H\subseteq I$ or if $c\leq\ddim$ are ordinals and
any hyperedge of $H$ has a subsequence in $I$,
since  in these cases, every $I$-independent subset of ${}^\kappa\kappa$ is $H$-independent.
Previously, we defined $I$ to be $H$-full if every $I$-independent subset of ${}^\kappa\kappa$ was also $H$-independent

Suppose $H$ and $I$ are relatively box-open $\ddim$- and $c$- dihypergraphs, respectively, on ${}^\kappa\kappa$.
If $H\leqf I$ and
there exists a
continuous homomorphism
from $\dhH \ddim$ to $H$,
then there exists a continuous homomorphism
from $\dhH  c$ to $I$.

Suppose that $f$ is a continuous homomorphism from $\dhH \ddim$ to $H$.
The next claim is needed in the case that $\domh{H}\not\subseteq \domh{I}$
to ensure that
$\ran([\iota])\subseteq \domh{I}$ for the map $\iota$ constructed below.

There is a sequence $\langle U_\alpha : \alpha<\kappa\rangle$ of open subsets of ${}^\kappa\kappa$ such that their intersection $U:=\bigcap_{\alpha<\kappa} U_\alpha$ satisfies
$U\cap\domh H\subseteq\domh I$ and
${I\restr(U\cap f(N_{e(t)}))}\neq\emptyset$ for all $t\in{}^\kappa \ddim$.

Since $A:=\domh H\setminus\domh I$ is $I$-independent and $H\leqf I$, $A$ is the union of $H$-independent sets $A_\alpha$ for $\alpha<\kappa$.
We show that the sets $U_\alpha:={}^\kappa\kappa\oldsetminus \closure{A}_\alpha$ are as required.
The first requirement is immediate.
Seeking a contradiction, suppose that $I\restr (U\cap f(N_t))=\emptyset$ for some $t\in{}^\kappa\ddim$.
Then $H\restr (U\cap f(N_t))$ admits a $\kappa$-coloring
since $H\leqf I$.
Moreover, $H\restr (f(N_t)\setminus U)$ also admits a $\kappa$-coloring, since $\closure A_\alpha$ is $H$-independent for each $\alpha<\kappa$ by Corollary <ref>.

This is the only place we need to use that $H$ is relatively box-open
Hence $H\restr f(N_t)$ admits a $\kappa$-coloring, contradicting Lemma <ref> <ref>.

We construct
continuous strict order preserving
$\iota:{}^{<\kappa} c\to{}^{<\kappa}\kappa$
$e:{}^{<\kappa} c\to{}^{<\kappa} \ddim$
such that for all $t\in{}^{<\kappa} c$ and $i\in c$,
\beginenumerate-(i)

$f(N_{e(t)})\subseteq N_{\iota(t)}$,

$\prod_{j\in  c}(N_{\iota(t\conc\langle j\rangle)}\cap\domh I)\subseteq I$, and

$N_{\iota(t\conc\langle i\rangle)}\subseteq U_{\lh(t)}$.
\endenumerate-(i)
$\iota(t)$ and $e(t)$ are constructed by recursion on $\lh(t)$
by modifying the construction for Lemma <ref> <ref> at successor lengths. In detail, let

Let $t\in{}^{<\kappa} c\,\setminus\{\emptyset\}$, and suppose that $\iota(u)$ and $e(u)$ have been defined for all $u\subsetneq t$.
If $\lh(t)\in\Lim$,
let $\iota(t):=\bigcup_{u\subsetneq t}\iota(u)$ and
$e(t):=\bigcup_{u\subsetneq t}e(u)$.
Then <ref> holds for
$t$ similarly to the proof of Lemma <ref> <ref>.

Now, suppose $\lh(t)\in\Succ$ and
let $u$ be the immediate predecessor of $t$.
$\iota(u\conc\langle i\rangle)$
$e(u\conc\langle i\rangle)$
for each $i\in  c$ simultaneously as follows.
Let $\bar x:=\langle x_i:i\in c\rangle$ be a sequence in $N_{e(u)}$
with $\langle f(x_i):i\in c\rangle\in I\restr U$.
Since $I$ is relatively box-open and $f(x_i)\in U\cap\domh I \subseteq U_{\lh(u)}\cap \domh I$ for each $i\in c$,
take a sequence
$\langle\iota(u\conc\langle i\rangle): i\in  c\rangle$
$\iota(u)\subsetneq\iota(u\conc\langle i\rangle)\subsetneq f(x_i)$
such that <ref> and <ref> hold for $u$.
By the continuity of $f$, choose
$e(u\conc\langle i\rangle)$ with $e(u)\subsetneq e(u\conc\langle i\rangle)\subsetneq x_{i}$ such that
<ref> holds for $u\conc\langle i\rangle$ for each $i\in c$.

Suppose that $\iota$ and $e$ have been constructed.
Then $[\iota]=f\comp[e]$

as in the proof of Lemma <ref> <ref>, so
$\ran([\iota])\subseteq\ran(f)\subseteq\domh H$. Moreover, $\ran([\iota])\subseteq U$ by <ref>, and hence $\ran([\iota])\subseteq \domh I$.

Thus $\iota$ is an order homomorphism for $I$ by <ref>,
so $[\iota]$ is a continuous homomorphism from $\dhH  c$ to $I$ by

Lemma <ref> <ref>.

Suppose $H$ and $I$ are relatively box-open $\ddim$- and $c$- dihypergraphs, respectively, on ${}^\kappa\kappa$.
If $H\equivf I$, then
[Equivalently, there exists a continuous homomorphism from $\dhH\ddim$ to $H$ if and only if there exists a continuous homomorphism from $\dhH c$ to $I$.]

Suppose that $H$ and $I$ are $\ddim$- and $c$- dihypergraphs, respectively, on ${}^\kappa\kappa$ and $I$ is box-open on some superset $X$ of $\domh H$.
\beginenumerate-(1)

The assumption that $H$ is relatively box-open is not needed for Lemma <ref>.
To see this, we modify its proof as follows.
The claim is not needed, since $\domh{H}\subseteq X$.
We construct $\iota$ and $e$ satisfying <ref> and <ref> with $\domh{I}$ replaced by $X$.
The construction works as above, except at successor lengths when $\iota(u)$ and $e(u)$ have been constructed,
it suffices to find a hyperedge
$\langle f(x_i): i\in c\rangle$ of $I\restr f(N_{e(u)})$.
This can be done since

$H\leqf I$ and
$H\restr f(N_t)$ does not admit a $\kappa$-coloring
by Lemma <ref> <ref>.
Since $I$ is box-open on $X$, choose $\iota(u\conc\langle i\rangle)\subsetneq f(x_i)$ extending $\iota(u)$ for each $i\in c$ so that
$\prod_{i\in  c}(N_{\iota(u\conc\langle i\rangle)}\cap X)\subseteq I$,

and choose each $e(u\conc\langle i\rangle)$
as above.
Then $[\iota]=f\comp[e]$
as in the proof of Lemma <ref> <ref>,
so $\ran([\iota])\subseteq \domh H\subseteq X$.
Therefore $\iota$ is an order homomorphism for $(X,I)$ and hence $[\iota]$ is a continuous homomorphism from $\dhH c$ to $I$.

If $c=\ddim$ and $I\subseteq H$,
then the assumption that $H$ is relatively box-open is not needed in the previous corollary.
This follows from <ref> and the fact that any homomorphism from $\dhH\ddim$ to $I$ is a homomorphism to $H$.
\endenumerate-(1)

By <ref>,
$\ODD\kappa\ddim(X)$ implies $\ODD\kappa{H}$
if $H$ contains a subdihypergraph $I$ which is box-open on $X$ and $H$-full, even if $H$ is not relatively box-open itself.

Suppose that $H$ and $I$ are $\ddim$- and $c$- dihypergraphs, respectively, on ${}^\kappa\kappa$.
We call $I$ injectively $H$-full ($H\leqif I$)
if for every hyperedge $\bar x=\langle x_j:j\in \ddim\rangle$ of $H$,
there exists an injective map $\pi:c\to\ddim$
with $\langle x_{\pi(i)}:i\in c\rangle\in I$.
[If $I$ is injectively $H$-full, then $I$ is $H$-full
since every $I$-independent subset of ${}^\kappa\kappa$ is also $H$-independent.]
We write $H\equivif I$ if $H\leqif I$ and $I \leqif H$.
If $I$ is injectively $H$-full, then $|c|\leq\|\ddim|$.
For instance, if $c\leq \ddim$ are ordinals and any hyperedge of $H$ has a subsequence in $I$,
then $I\leqif H$.

If $I\leqif H$ and $I$ is box-open on a set $X$ with $\domh H\subseteq X$, then we can assume that the continuous homomorphism from $\dhH c$ to $I$ in  Lemma <ref>
is of the form $f\comp [e]$
for some continuous strict $\wedge$-homomorphism $e:{}^{<\kappa} c\to{}^{<\kappa}\ddim$ and some continuous homomorphism $f$ from $\dhH\ddim$ to $H$.
Moreover, we do not need to assume that $H$ is relatively box-open.

To see this, we modify the construction in the previous remark at successor lengths as follows.
Assuming $\iota(u)$ and $e(u)$ have been constructed,

$x_j\in{}^{\kappa} \ddim$ extend $e(u)\conc\langle j\rangle$ for each $j\in \ddim$.
Since $\langle f(x_j):j\in  \ddim\rangle\in H$ and $I\leqif H$,
$\langle f(x_{\pi_u(i)}):i\in  c\rangle$ is in $I$ for some injective map $\pi_u:c\to\ddim$.
Since $I$ is box-open on $X$, take a sequence
$\langle\iota(u\conc\langle i\rangle): i\in  c\rangle$
such that
$\iota(u)\subsetneq\iota(u\conc\langle i\rangle)\subsetneq f(x_{\pi_u(i)})$
and $\prod_{j\in  c}(N_{\iota(t\conc\langle j\rangle)}\cap X)\subseteq I$.

By the continuity of $f$, choose
$e(u\conc\langle i\rangle)$
$e(u)\conc\langle\pi_u(i)\rangle\subseteq e(u\conc\langle i\rangle)\subsetneq x_{\pi_u(i)}$
which also satisfies <ref> from the previous construction.

As above, $\iota$ is an order homomorphism for $(X,I)$ with $[\iota]=f\circ [e]$.
Since $\pi_u$ is injective for all $u\in {}^{<\kappa}\kappa$,
$e(u\conc\langle i\rangle)\wedge e(u\conc\langle j\rangle)=e(u)$ for all $i\neq j\in c$, so $e$ is a strict $\wedge$-homomorphism.

It follows that in this situation, $[e]$ is a homeomorphism onto its image.
Thus, if $f$ is injective or a homeomorphism onto its image, then the same holds for $f\comp [e]$. This will be relevant in  Subsection <ref>.

If $H$ and $I$ are $\ddim$-dihypergraphs with $H\subseteq I$,
then any $\kappa$-coloring of $I$ is a $\kappa$-coloring of $H$, and any continuous homomorphism from $\dhH\ddim$ to $H$ is one to $I$ as well.
Then next lemma extends this observation to box-open dihypergraphs $I$ with the following properties.
In the next definition, lemma and corollary, we assume that
$c$ is a discrete topological space with $|c|\geq 2$.

Suppose that $H$ is a $\ddim$-dihypergraph and $I$ is a $c$-dihypergraph on ${}^\kappa\kappa$.
$I$ is $H$-full if
$I\restr\ran(\bar x)$ is nonempty for every hyperedge $\bar x$ of $H$.

Note that
$I$ is $H$-full if and only if every $I$-independent subset of ${}^\kappa\kappa$ is $H$-independent.
If $c\leq \ddim$ are ordinals and any hyperedge of $H$ has a subsequence in $I$,
then $I$ is $H$-full.

Suppose that $X\subseteq{}^\kappa\kappa$ and $H$ and $I$ are $\ddim$- and $c$-dihypergraphs on $X$, respectively.

Suppose that $I$ is $H$-full.
\beginenumerate-(1)

Any $\kappa$-coloring of $I$ is a $\kappa$-coloring of $H$.

If $I$ is box-open on $X$ and
there exists a
continuous homomorphism
from $\dhH \ddim$ to $H$,
then there exists a continuous homomorphism
from $\dhH  c$ to $I$.
\endenumerate-(1)

<ref> holds since
any $I$-independent subset of $X$ is $H$-independent.
The argument for <ref> is similar to the proof of Lemma <ref> <ref>.
Let $f$ be a continuous homomorphism from $\dhH \ddim$ to $H$.
We construct
continuous strict order preserving
$\iota:{}^{<\kappa} c\to{}^{<\kappa}\kappa$
$e:{}^{<\kappa} c\to{}^{<\kappa} \ddim$
such that for all $t\in{}^{<\kappa} c$,
\beginenumerate-(i)

$f(N_{e(t)})\subseteq N_{\iota(t)}$, and

$\prod_{i\in  c}(N_{\iota(t\conc\langle i\rangle)}\cap X)\subseteq I$.
\endenumerate-(i)
$\iota(t)$ and $e(t)$ are constructed by recursion on $\lh(t)$
by modifying the construction for Lemma <ref> <ref> at successor lengths.
In detail,
let $\iota(\emptyset):=e(\emptyset)=\emptyset$.
Let $t\in{}^{<\kappa}  c\,\setminus\{\emptyset\}$, and suppose that $\iota(u)$ and $e(u)$ have been defined for all $u\subsetneq t$.
If $\lh(t)\in\Lim$,
let $\iota(t):=\bigcup_{u\subsetneq t}\iota(u)$ and
$e(t):=\bigcup_{u\subsetneq t}e(u)$.
Then <ref> holds for
$t$ similarly to the proof of Lemma <ref> <ref>.

Now, suppose $\lh(t)\in\Succ$ and
let $u$ be the immediate predecessor of $t$.
$\iota(u\conc\langle i\rangle)$
$e(u\conc\langle i\rangle)$
for each $i\in  c$ simultaneously as follows.
$z_j\in{}^{\kappa} \ddim$ extend $e(u)\conc\langle j\rangle$ for each $j\in \ddim$.
Since $\langle f(z_j):j\in  \ddim\rangle\in H$ and $I$ is $H$-full,
there exists $\pi:=\pi_u\in{}^c\ddim$
with $\langle f(z_{\pi(i)}):i\in  c\rangle\in I$.
Since $I$ is box-open on $X$, take a sequence
$\langle\iota(u\conc\langle i\rangle): i\in  c\rangle$
such that
$\iota(u)\subsetneq\iota(u\conc\langle i\rangle)\subsetneq f(z_{\pi(i)})$
and <ref> holds for $u$.
By the continuity of $f$, choose
$e(u\conc\langle i\rangle)$ with $e(u)\conc\langle\pi(i)\rangle\subseteq e(u\conc\langle i\rangle)\subsetneq z_{\pi(i)}$ such that
<ref> holds for $u\conc\langle i\rangle$ for each $i\in c$.

Suppose that $\iota$ and $e$ have been constructed.
Then $g:=[\iota]$ is equal to $f\comp[e]$
as in the proof of Lemma <ref> <ref>.
Hence $\iota$ is an order homomorphism for $(X,I)$ by <ref>,
and $g$ is a continuous homomorphism from $\dhH  c$ to $I$ by
Lemma <ref> <ref>.

Suppose that $X\subseteq{}^\kappa\kappa$ and $H$ and $I$ are $\ddim$- and $c$-dihypergraphs on $X$, respectively.

Suppose that $I$ is $H$-full and $H$ is $I$-full.
\beginenumerate-(1)

A map $f:X\to\kappa$ is a $\kappa$-coloring of $H$ if and only if it is a $\kappa$-coloring of $I$.

If $H$ and $I$ are box-open on $X$, then
there exists a continuous homomorphism from $\dhH\ddim$ to $H$ if and only if there exists a continuous homomorphism from $\dhH c$ to $I$.
\endenumerate-(1)
Thus, $\ODD\kappa{H}$ $\Leftrightarrow$ $\ODD\kappa{I}$ if $H$ and $I$ are box-open on $X$.

Note that the assumption in this corollary is equivalent to
the statement that
any subset of ${}^\kappa\kappa$ is $H$-independent if and only if it is $I$-independent.
Moreover, if $c=\ddim$ and $H\subseteq I$,
then the assumption that $I$ is box-open is not needed for <ref> and the last claim in the corollary.
$\ODD\kappa\ddim(X)$ implies $\ODD\kappa{I}$
if $I$ contains a subdihypergraph $H$ which is box-open on $X$ and $I$-full, even if $I$ is not box-open itself.

that $H$ and $I$ are $\ddim$- and $c$-dihypergraphs on a subset $X$ of ${}^\kappa\kappa$, respectively.
We call $I$ injectively $H$-full in analogy with Definition <ref>
if for every hyperedge $\bar x=\langle x_j:j\in \ddim\rangle$ of $H$,
there exists an injective map $\pi:c\to\ddim$
with $\langle x_{\pi(i)}:i\in c\rangle\in I$.
[If $I$ is injectively $H$-full, then $I$ is clearly $H$-full and $|c|\leq|\ddim|$.]
For instance, if $c\leq \ddim$ are ordinals and any hyperedge of $H$ has a subsequence in $I$,
then $I$ is injectively $H$-full.
If $I$ is injectively $H$-full, then we can assume that $g=f\comp [e]$ for some continuous strict $\wedge$-homomorphism $e:{}^{<\kappa} c\to{}^{<\kappa}\ddim$ in Lemma <ref> <ref>.
To see this,
we modify the construction in the previous proof at successor stages
by choosing the maps $\pi_u$ to be injective for each $u\in{}^{<\kappa}c$.
$e(u\conc\langle i\rangle)\wedge e(u\conc\langle j\rangle)=e(u)$ for all $i\neq j\in c$, so $e$ is a strict $\wedge$-homomorphism.
It follows that $[e]$ is a homeomorphism onto its image.
Thus if $f$ is injective or a homeomorphism onto its image, the same holds for $g=f\comp [e]$. This will be relevant in  Subsection <ref>.

§ THE OPEN DIHYPERGRAPH DICHOTOMY FOR DEFINABLE SETS

In this section we prove the main result, Theorem <ref>.
The argument consists of three steps.
In the situation of the theorem,
$\kappa$ is a regular infinite cardinal,
$2\leq\ddim\leq \kappa$, and
$X$ is a subset of ${}^\kappa\kappa$ and
$H$ is a box-open $\ddim$-dihypergraph on ${}^\kappa\kappa$
in some $\Col(\kappa,\lle\lambda)$-generic extension $V[G]$.
We may assume that $H\restr X$ does not admit a $\kappa$-coloring.
In the first step, we find an ordinal $\nu<\lambda$ that reflects the set $\mathcal T$ of trees that code $H$-independent closed sets
in the sense that
${\mathcal T}\cap V[G_\nu] \in V[G_\nu]$,
$G_\nu$ denotes the restriction of $G$ to $\Col(\kappa,\lle\nu)$.

In the second step,
we work in $V[G_\nu]$
to find a $\Col(\kappa,\lle\lambda)$-name
for an element of $X$
that avoids all closed sets induced by trees in $\mathcal T \cap V[G_\nu]$.
Then we
transform this to an $\Add(\kappa,1)$-name $\nna$ to facilitate the third step,
where $\nna$ is used
to construct a continuous homomorphism
from $\dhH\ddim$ to $H\restr X$.
The values of this homomorphism come from the evaluations of $\nna$
by a $\kappa$-perfect set of
filters with well-behaved quotient forcings.
The main technical step in the proof, the Nice Quotient Lemma for $\Col(\kappa,\lle\lambda)$,

guarantees the existence
in $V[G]$
of such a $\kappa$-perfect set.
More precisely, it constructs
a continuous function $g$
from ${}^\kappa\ddim$
to the set of $\Add(\kappa,1)$-generic filters with well-behaved quotient forcings
such that $g$ induces a

homomorphism from $\dhH\ddim$ to $H$ via $\nna$.
Controlling the quotient forcings is necessary to ensure the values of the homomorphism are in $X$.

The proof of the Nice Quotient Lemma is postponed to Subsections <ref> and <ref>.

In the countable case,
our proof of the main theorem
Solovay's proof of the consistency of  the perfect set property $\PSP_\omega(\defsets\omega)$ <cit.> and
Feng's proof of the consistency of the open graph dichotomy $\OGD_\omega(\defsets\omega)$ <cit.>.
[See Definitions <ref> and <ref>.]
In this case, the Nice Quotient Lemma is an easy consequence of Solovay's lemma showing that all elements of the Baire space in $V[G]$ have $\Col(\omega,\lle\lambda)$ as a quotient.
However, it is well known that Solovay's lemma fails in the uncountable case.
The Nice Quotient Lemma overcomes this difficulty by ensuring that
all elements of the $\kappa$-Baire space
that occur in the construction do have $\Col(\kappa,\lle\lambda)$ as a quotient in the uncountable case.
This was done in <cit.>
for the $\kappa$-perfect set property using a similar idea but different technical means.

§.§ Three important steps

§.§.§ Reflection to intermediate models

We will work with the Lévy collapse of an inaccessible or a
Mahlo cardinal.
[A cardinal $\lambda$ is Mahlo if the set of inaccessible cardinals $\nu<\lambda$ is stationary in $\lambda$.]
The next lemma describes an essential property of such extensions.

Recall the notation $\PP_\alpha, \PP_I, \PP^\alpha, G_\alpha, G_I, G^\alpha$ from page notation for the Levy collapse and subforcings.
Note that if $\lambda$ is inaccessible, then $\PP^\alpha$ is equivalent to $\PP_\lambda$ for all $\alpha<\lambda$, by Corollary <ref>.

Suppose that $\kappa<\lambda$ are regular infinite cardinals, $G$ is $\PP_\lambda$-generic over $V$ and $X$ is a subset of ${}^\kappa\kappa$ in $V[G]$.
Suppose that either
\beginenumerate-(1)

$\lambda$ is inaccessible and $X\in \defsets\kappa^{V[G]}$,

$\lambda$ is Mahlo.
\endenumerate-(1)
Then there is a stationary (in $V$) subset $S\in V$ of $\lambda$ such that for all $\nu\in S$,  we have $X \cap V[G_\nu] \in V[G_\nu]$.
In fact, assuming <ref>, $S$ can be chosen as a final segment of $\lambda$.
Assuming <ref>, $S$ can be chosen so that every $\nu\in S$ is inaccessible.

First assume <ref>.
In $V[G]$, let

$$X:=X_{\varphi,a}:=\{ x\in {}^\kappa\kappa: \varphi(x,a) \},$$

where $\varphi(x,a)$ is a formula with a parameter $a\in {}^\kappa\Ord$.
Since $\PP_\lambda$ has the $\lambda$-c.c., $a\in V[G_\alpha]$ for some $\alpha<\lambda$.
We shall show that $X\cap V[G_\nu]\in V[G_\nu]$ for any $\nu\in S:=[\alpha,\lambda)$.
Let $x\in({}^\kappa\kappa)^{V[G_\nu]}$.
Since $a$ is also in $V[G_\nu]$ and
$V[G]$ is a $\PP^\nu$-generic extension of $V[G_\nu]$ and
$\PP^\nu$ is homogeneous,

\[
x\in X=X_{\varphi,a}
\ \ \Longleftrightarrow \ \
\one_{\PP^\nu}\forces^{V[G_\nu]} \varphi(x,a).
\]

$X\cap V[G_\nu]$ is equal to the set $\{ x\in({}^\kappa\kappa)^{V[G_\nu]}:
\one_{\PP^\nu}\forces^{V[G_\nu]} \varphi(x,a)\}$,
which is in $V[G_\nu]$.

Now assume <ref>.
Let $\dot{X}\in V$ be a $\PP_\lambda$-name with $\dot{X}^G=X$.

Define $f: \lambda\to \lambda$ as follows.
Given $\alpha<\lambda$, let $\mathrm{Name}_\alpha$ denote the set of nice $\PP_\alpha$-names for subsets of $\kappa\times \kappa$.
Note that $|\mathrm{Name}_\alpha |\leq2^{\max\{\kappa,|\alpha|\}}$.
For each $\dot{x}\in \mathrm{Name}_\alpha$, choose a maximal antichain $A_{\dot{x}}$ in $\PP_\lambda$ of conditions deciding whether $\dot{x}\in \dot{X}$.
By the $\lambda$-c.c. of $\PP_\lambda$,
let $\mu_{\dot{x}}<\lambda$ be least with $A_{\dot{x}}\subseteq \PP_{\mu_{\dot{x}}}$.
Finally, let $f(\alpha):=\sup \{ \mu_{\dot{x}}: \dot{x}\in \mathrm{Name}_\alpha\}<\lambda$.

In $V$, there is a club of closure points of $f$, i.e. ordinals $\nu<\lambda$ with  $f(\alpha)<\nu$ for all $\alpha<\nu$.

Since $\lambda$ is Mahlo, the set $S$ of inaccessible closure points of $f$ is stationary in $\lambda$.

Let $\nu\in S$.
We shall show that $X\cap V[G_\nu] \in V[G_\nu]$.

For any nice $\PP_\nu$-name $\dot{x}$ for a subset of $\kappa\times\kappa$, exactly one of the following holds:
\beginenumerate-(i)

* $\exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \dot{x} \in \dot{X}$.

* $\exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \dot{x} \notin \dot{X}$.
\endenumerate-(i)

Since $\nu$ is inaccessible, $\PP_\nu$ has the $\nu$-c.c.
Since $\dot{x}$ is a nice $\PP_\nu$-name, it is a $\PP_\gamma$-name for some $\gamma<\nu$.
By the definition of $f$, there is
$A\subseteq \PP_{f(\gamma)}$ such that $A$ is
a maximal antichain in $\PP_\lambda$ of conditions deciding whether $\dot{x}\in \dot{X}$.
Since $\nu$ is a closure point of $f$, we have $f(\gamma)<\nu$.
Thus $A$ is a maximal antichain in $\PP_\nu$.
Let $p\in G_\nu \cap A$.

If $\dot{x}\in \mathrm{Name}_\alpha$ and $\dot{x}^{G_\nu}=x$, then:
\beginenumerate-(i)

* $x\in X \ \Longleftrightarrow\  \exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \dot{x} \in \dot{X}$.

* $x\notin X \ \Longleftrightarrow\ \exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \dot{x} \notin \dot{X}$.
\endenumerate-(i)

By the previous claim, it suffices to show the implications from right to left.
Suppose $p\in G_\nu$ and  $p\forces_{\PP_\lambda}^V \dot{x} \in \dot{X}$.
Since $\dot{x}$ is a $\PP_\nu$-name and $G_\nu\subseteq G$, we have $\dot{x}^{G_\nu} = \dot{x}^G\in \dot{X}^G$.
The second implication is similar.

In $V[G_\nu]$, define
$X_\nu:=\{ {\dot x}^{G_\nu}:\: \exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \dot{x} \in \dot{X} \}$.
Then we have $X\cap V[G_\nu]=X_\nu \in V[G_\nu]$.

The previous lemma also has a version for functions:
Let $\kappa<\lambda$ be regular infinite cardinals, let $G$ be $\PP_\lambda$-generic over $V$ and let $F: {}^\kappa\kappa\to {}^\kappa\kappa$ be a function in $V[G]$.
If either
\beginenumerate-(1)

$\lambda$ is inaccessible and $F\in \defsets\kappa^{V[G]}$, or

$\lambda$ is Mahlo,
\endenumerate-(1)
then there is a stationary (in $V$) subset $S\in V$ of $\lambda$ such that for all $\nu\in S$,  we have $F\restr V[G_\nu] \in V[G_\nu]$.
Assuming <ref>, $S$ can be chosen as a final segment of $\lambda$, and for <ref>, $S$ can be chosen so that every $\nu\in S$ is inaccessible.

To see this, apply the previous lemma

to the set

$X= \{\langle \alpha,\beta \rangle\conc x : x\in {}^\kappa\kappa,\, (\alpha,\beta)\in F(x) \}$.
If $X \cap V[G_\nu] \in V[G_\nu]$, then $F\restr V[G_\nu] \in V[G_\nu]$,
If $x\in{}^\kappa\kappa\cap V[G_\nu]$, then $F(x)=\{(\alpha,\beta): \langle \alpha,\beta\rangle\conc x\in X\cap V[G_\nu]$
since $F\restr V[G_\nu]$ can be calculated from $X \cap V[G_\nu]$.

The Mahlo cardinal is necessary in Lemma <ref> <ref> and Remark <ref> <ref>.
To see this, assume the $\GCH$ holds and $\lambda>\kappa$ is any cardinal that is not Mahlo.
We claim that
there exists a function $F:{}^\kappa\kappa\to{}^\kappa\kappa$ in $V[G]$ such that
$F\restr  V[G_\alpha]\notin V[G_\alpha]$ for all $\alpha$ with $\kappa\leq\alpha<\lambda$.

First suppose $\lambda$ is singular in $V$.
Then $\lambda$ remains singular in $V[G]$.
Hence $\lambda\neq (\kappa^+)^{V[G]}$ and $\lambda$ is collapsed in $V[G]$, so there exists some $x\in ({}^\kappa\kappa)^{V[G]}$ with $x\notin V[G_\alpha]$ for all $\alpha<\lambda$.
Then any $F$ mapping $\langle 0\rangle^\kappa$ to $x$ works

Now suppose $\lambda$ is regular in $V$.
Let $C$ be a club in $\lambda$ whose elements are not inaccessible.
Additionally, if $\lambda$ is a successor cardinal then assume $C$ does not contain cardinals, while if $\lambda$ is a limit cardinal then assume $C$  contains only cardinals.
For any $\alpha<\lambda$, let $\alpha'$ denote the least ordinal
above $\alpha$ in $C$.
Take any function $F: {}^\kappa\kappa\to {}^\kappa\kappa$ in $V[G]$ with the following property: if $\alpha<\lambda$ is least with $x\in V[G_\alpha]$, then $F(x)\notin V[G_{\alpha'}]$.

We show that
$F\restr  V[G_\alpha]\notin V[G_\alpha]$ for all $\alpha$ with $\kappa\leq\alpha<\lambda$.
To see this, fix any such $\alpha$.
We distinguish two cases:

$\alpha$ is not a limit point of $C$.

Then there is some $\gamma\in C$ with $\gamma< \alpha\leq \gamma'$.
Take any $x\in V[G_\alpha]\oldsetminus V[G_\gamma]$.
Let $\beta>\gamma$ be least with $x\in V[G_\beta]$.
Then $F(x)\notin V[G_{\beta'}]$ by the definition of $F$.
Since $\gamma<\beta\leq\alpha\leq\gamma'$, we have $\gamma'=\beta'=\alpha'$. But we only need $\alpha\leq\beta'$.
$\alpha\leq\gamma'\leq\beta'$, we have $F(x)\notin V[G_\alpha]$ as required.

$\alpha$ is a limit point of $C$.

We claim that there is some $x\in {}^\kappa\kappa$ such that $\alpha$ is least with $x\in V[G_\alpha]$.
First suppose that $\lambda$ is successor cardinal in $V$.[For successor cardinals $\lambda$, the proof does not use that $\alpha$ is a limit point of $C$, so Case <ref> is not needed.]
Let $\mu:=|\alpha|^V$, so $\mu < \alpha <(\mu^+)^V$
since $C$ does not contain cardinals.
Since $|\alpha|=\kappa$ in $V[G_{\mu+1}]$, $G_{(\mu+1,\alpha)}$ can be coded as a subset $x$ of $\kappa$ via a bijection $f: \kappa\to \alpha$ with $f\in V[G_{\mu+1}]$.
Then $\alpha$ is least with $x\in V[G_\alpha]$.
Now suppose $\lambda$ is a limit cardinal in $V$.
Since $C$ contains only cardinals and the $\GCH$ holds, $\alpha$ is singular in $V$.
Thus $\alpha\neq (\kappa^+)^{V[G_\alpha]}$ and $\alpha$ is collapsed in $V[G_\alpha]$, so the claim follows.

Then $F(x)\notin V[G_{\alpha'}]$ by the definition of $F$. Thus, $F(x)\notin V[G_\alpha]$ as required.

The proof of Lemma <ref> <ref>
can be adapted to show that if $\lambda>\kappa$ is inaccessible, then
the following strengthening of $\Diamond_{\kappa^+}$
holds in
$\PP_\lambda$-generic extensions $V[G]$:

$\Diamond^+_{\kappa^+}$: there exists a sequence
$\langle \mathcal A_\alpha:\alpha<\kappa^+\rangle$ of sets
$\mathcal A_\alpha\subseteq\pwrset(\alpha)$ such that
\beginenumerate-(a)

$|\mathcal A_\alpha|\leq \kappa$ for all $\alpha<\kappa^+$, and

for all $X\subseteq\kappa^+$, there exists a club $C$ in $\kappa^+$ such that for all $\alpha\in C$,
$X\cap\alpha \in \mathcal A_\alpha$ and $C\cap \alpha \in \mathcal A_\alpha$.
\endenumerate-(a)

We show that in $V[G]$,
$\langle\mathcal A_\alpha=\pwrset(\alpha)^{V[G_\alpha]}:\alpha<\kappa\rangle$
is a $\Diamond^+_{\kappa^+}$-sequence.
It is clear that <ref> holds in $V[G]$.

To show <ref>,
we use an argument similar to the proof of Lemma <ref> <ref>.
[Note that this version of the argument works assuming only that $\lambda$ is inaccessible.]

Let $X$ be a subset of $\kappa^+=\lambda$ in $V[G]$, and let $\dot{X}\in V$ be a $\PP_\lambda$-name with $\dot{X}^G=X$.

In $V$, first define a function $f: \lambda\to \lambda$ as follows.
For each $\alpha<\lambda$,
let $A$ be a maximal antichain in $\PP_\lambda$ consisting of conditions
deciding whether $\alpha\in\dot X$.
Since $\PP_\lambda$ has the $\lambda$-c.c., choose $f(\alpha)<\lambda$ with $A\subseteq \PP_{f(\alpha)}$.

Let $C$ be a club of limit ordinals $\nu<\lambda$ that are closure points of $f$.
We shall show that $C$ satisfies <ref>.
$\nu$ will not need to be inaccessible. So $\lambda$ does not need to be Mahlo
Since $C\in V$, it suffices to show that $X
\cap \nu \in \mathcal{A}_\nu=\mathcal{P}(\nu)^{V[G_\nu]}$ for any given $\nu\in C$.

For any $\alpha<\nu$, exactly one of the following holds:
\beginenumerate-(i)

* $\exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \alpha \in \dot{X}$.

* $\exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \alpha \notin \dot{X}$.
\endenumerate-(i)

By the definition of $f$, there is
$A\subseteq \PP_{f(\gamma)}$ such that $A$ is
a maximal antichain in $\PP_\lambda$ of conditions deciding whether $\alpha\in \dot{X}$.
Since $\nu$ is a closure point of $f$, we have $f(\gamma)<\nu$.
Thus $A$ is a maximal antichain in $\PP_\nu$.
Let $p\in G_\nu \cap A$.

As in the proof of Lemma <ref> <ref>,
it now follows that
for all $\alpha<\nu$,

$$\alpha\in X\cap\nu \  \Longleftrightarrow \
\exists p\in G_\nu\ \,p\forces_{\PP_\lambda}^V \alpha \in \dot{X}.$$

$X\cap\nu$ is an element of

§.§.§ Names and witness functions

The following assumptions will be used throughout the rest of this section, unless explicitly otherwise stated.

In $V$:

* $\kappa$ is a regular infinite cardinal,
and $\lambda>\kappa$ is an inaccessible cardinal.
$G$ is a $\PP_\lambda=\Col(\kappa,\lle\lambda)$-generic filter over $V$.

* $\kappa^{<\kappa}=\kappa$ holds in $V$.
This may be assumed without loss of generality,
because $\PP_2$ forces $\kappa{}^{<\kappa}=\kappa$ and $\PP^2$ is equivalent to $\PP_\lambda$.

In $V[G]$:

* $X$ is a subset of ${}^\kappa\kappa$ in $\defsetsk$.

More specifically

\[X=\Xphia=
\{x\in{}^{\kappa}\kappa:\varphi(x,a)\},\]

is a first order formula
$\kappa$ and $\ddim$ are clear from the context, so they do not need to be specified here
with a parameter $a\in{}^\kappa\Ord$.

* $H$ is a  box-open $\ddim$-dihypergraph on ${}^\kappa\kappa$
such that $H\restr X$ does not admit a $\kappa$-coloring.

* $\lambda$ is a Mahlo cardinal in $V$, or $H\in\defsetsk$ in $V[G]$.
[This assumption can be weakened as in Remark <ref> below.]
[If $\ddim<\kappa$, then $H\in\defsetsk$ follows from the assumption that $H$ is box-open by Lemma <ref>.]

To prove Theorem <ref>, it suffices to prove
that $\ODD\kappa{H\restr X}$ holds under these assumptions.
Since we assume that $H\restr X$ does not admit a $\kappa$-coloring,
it further suffices to show that there is
a continuous homomorphism from $\dhH\ddim$ to $H\restr X$.

We work in $V[G]$ unless otherwise stated. Let $T$ be a subtree of ${}^{<\kappa}\kappa$.
Recall from Definition <ref>
that $T$ is an $H$-independent tree if
for all sequences $\langle t_\alpha\in T:\alpha<\ddim\rangle$,
we have
$\prod_{\alpha<\ddim}N_{t_\alpha}\not\subseteq H$.
Since $H$ is box-open on ${}^\kappa\kappa$,
$[T]$ is an $H$-independent set
whenever $T$ is an $H$-independent tree
and the converse holds when $T$ is pruned,
by Lemma <ref>.

For all $\alpha\leq\lambda$, let
use the command “indtrees” for the notation $\indtrees\alpha$

\begin{align*}
\mathcal T_\alpha &:=\{T\in V[G_\alpha] : T
\text{ is a subtree of ${}^{<\kappa}\kappa$}\},
\\
\indtrees\alpha &:=\{T\in V[G_\alpha] : T
\text{ is an $H$-independent subtree of ${}^{<\kappa}\kappa$}\}.
\end{align*}

Let $\alpha<\lambda$.
Then $\mathcal T_\alpha$ has size $\kappa$ in $V[G]$
since $\lambda$ is inaccessible,
and $\mathcal T_\alpha\in V[G_\alpha]$.
However, $\indtrees\alpha$ may not be in $V[G_\alpha]$.

There is a stationary (in $V$) subset $S\in V$ of $\lambda$ such that
$\indtrees\nu\in V[G_\nu]$ for all $\nu\in S$.

In $V[G]$,
code $\indtrees\lambda\subseteq\pwrset({}^{<\kappa}\kappa)$ as a subset $Y_\lambda$ of ${}^\kappa\kappa$, using a bijection $f: \kappa\to {}^{<\kappa}\kappa$ in $V$.
If $H\in\defsetsk$, then $\indtrees\lambda\in\defsetsk$ and hence $Y_\lambda\in\defsetsk$.
Otherwise, $\lambda$ is Mahlo.
In both cases, Lemma <ref> yields stationarily many cardinals $\nu<\lambda$  such that $Y_\lambda\cap V[G_\nu]\in V[G_\nu]$, and hence $\indtrees\nu=\indtrees\lambda\cap V[G_\nu]\in V[G_\nu]$.

For the rest of this section, we fix a cardinal $\nu$
such that $\indtrees\nu\in V[G_\nu]$ and $a\in V[G_\nu]$,
where $a\in({}^\kappa\Ord)^{V[G]}$ is the parameter with $X=\Xphia^{V[G]}$.
This may be done by the previous lemma and the $\lambda$-c.c. for $\PP_\lambda$.

Note that the assumption that either $H\in \defsetsk^{V[G]}$ or $\lambda$ is Mahlo in $V$ in Assumption <ref>
was needed only to obtain the previous lemma.
From now on,
we will use only the fact that
$\indtrees\nu\in V[G_\nu]$, $a\in V[G_\nu]$ and the rest of Assumption <ref>.

The next lemma shows that there is a $\PP_\lambda$-name $\na$ in $V[G_\nu]$ for an element
of $X$ that
avoids all sets $[T]$ with $T\in\indtrees\nu$.

In $V[G_\nu]$, there exists a $\PP_\lambda$-name $\na$
with the following properties:
\beginenumerate-(1)

$\one_{\PP_\lambda} \forces^{V[G_\nu]}{\na}\in\Xphia\oldsetminus V[G_\nu].$
[Note that
$\one_{\PP_\lambda} \forces^{V[G_\nu]}{\na}\notin V[G_\nu]$ follows from <ref> as in the proof of the lemma.]

$\one_{\PP_\lambda}\forces^{V[G_\nu]} {\na\notin[T]}$ for all ${T\in\indtrees\nu}$.
\endenumerate-(1)

Work in $V[G]$.
$[T]$ is an $H$-independent set for all $T\in\indtrees\nu$
by Lemma <ref> <ref>.
$H\restr X=H\restr\Xphia$ does not admit a $\kappa$-coloring and
$\indtrees\nu$ has size at most $\kappa$,

we have

\begin{equation}
\label{3a eq}
\Xphia-\bigcup\big\{[T]: T\in\indtrees\nu\big\}
\neq\emptyset.
\end{equation}

Note that (<ref>)
is a first-order statement with parameters in $V[G_\nu]$,
because $\nu$ was chosen so that
$\indtrees\nu$ and $a$ are in $V[G_\nu]$.

This will be the only place
in the proof of Theorem <ref>
where we use
Lemma <ref>.
Since $\PP^\nu$ is homogeneous,
$\one_{\PP^\nu}$ forces in
$V[G_\nu]$ that (<ref>) holds.
Since the forcings $\PP^\nu$ and $\PP_\lambda$ are equivalent
by Corollary <ref>,
$\one_{\PP_\lambda}$ forces in
$V[G_\nu]$ that (<ref>) holds.
By the maximal principle, there exists a $\PP_\lambda$-name $\na\in V[G_\nu]$
for an element of $\Xphia$ which satisfies <ref>.
To see that
$\one_{\PP_\lambda}\forces^{V[G_\nu]} {\na\notin V[G_\nu]}$,
let $x\in({}^\kappa\kappa)^{V[G_\nu]}$.
Since $\{x\}$ is $H$-independent,
is in $\indtrees\nu$ by Lemma <ref> <ref>.
Thus $\one_{\PP_\lambda}\forces^{V[G_\nu]} \sigma\neq x$ by <ref>.

Recall from Definition <ref> that
if $\QQ$ is a forcing, $q\in\QQ$ and $\nna$ is a $\QQ$-name for an element of ${}^\kappa\kappa$,

$$T^{\nna, q}=T^{\nna, q}_\QQ= \{t\in{}^{<\kappa}\kappa\::\: \exists r\leq q\ \ r\forces t\subseteq \nna\}$$

is the tree of possible values for $\nna$ below $q$.
Recall also that the definition of $T^{\nna,q}$ is absolute between transitive models of $\ZFC$ with the same ${}^{<\kappa}\kappa$
by Lemma <ref> <ref>.
In particular,
if $\nna\in V[G_\nu]$, then

The next lemma shows that the trees
$T^{\na,p}_{\PP_\lambda}$ are not $H$-independent for names $\na$ as in the previous lemma.

$T^{\na,p}_{\PP_\lambda}\in\mathcal T_\nu\oldsetminus\indtrees\nu$
for all names $\na$ as in Lemma <ref> and

all $p\in\PP_\lambda$.

Let $p\in \PP_\lambda$, and let
Since $\na\in V[G_\nu]$, we have
$T\in \mathcal T_\nu$ by Lemma <ref> <ref>.
Moreover, $p\forces^{V[G_\nu]}_{\PP_\lambda}\na\in[T]$ by Lemma <ref> <ref>.
$T\notin \indtrees\nu$
since $\na$ was chosen so that Lemma <ref> <ref> holds.

We now prove an analogue of the two previous lemmas for $\Add(\kappa,1)$.
Recall that

\[\Add(\kappa,1):=\{p:\alpha\to\kappa\,\mid\, \alpha<\kappa\}.%
\footnote{See Definition~\ref{definition of Add(kappa,1)}.}
\]

For unboundedly many $\gamma<\lambda$, there is an $\Add(\kappa,1)$-name $\nna$ in $V[G_\gamma]$
for a new element of ${}^\kappa\kappa$
with the following properties:
\beginenumerate-(1)

In $V[G_\gamma]$,
$\one_{\Add(\kappa,1)}$ forces that

$T^{\,\nna,\,q}_{\Add(\kappa,1)}\in\mathcal T_\gamma\oldsetminus\indtrees\gamma$
for all $q\in\Add(\kappa,1)$.
\endenumerate-(1)

Fix a $\PP_\lambda$-name $\na$ as in Lemma <ref>.
We first sketch the idea.
First choose some $\gamma$ so that
$\na\in V[G_\gamma]$ is a $\PP_\gamma$-name
and the forcings $\PP_\gamma$ and
$\Add(\kappa,1)$ are equivalent in $V[G_\gamma]$.
Then transform
to an $\Add(\kappa,1)$-name $\nna\in V[G_\gamma]$,
and finally show that
Lemma <ref> <ref>