## Annotated Content

$\mathchar 632\relax$0 Introduction

$\mathchar 632\relax$1 Middle Diamond: sufficient conditions

$\mathchar 632\relax$2 Upgrading to large cardinals

$\mathchar 632\relax$3 Catching more cofinalities

$\mathchar 632\relax$4 Further upgrading

$\mathchar 632\relax$5 Relative of $I[\lambda]$

$\mathchar 632\relax$6 Relative of $\clubsuit$

? EK ?

##  $\mathchar 632\relax$0 Introduction

Using the diamond (discovered by Jensen [ J]) we can construct many objects;
wonderful but there is a price - we must assume $\mathbb{V}=\mathbb{L}$ or at
least instances of G.C.H. (see [ Sh:460] and citations there).

? J ?

? Sh:460 ?

Naturally we would like to get the benefit without paying the price. A way to
do this is to use the weak diamond (see [ DvSh:65], [ Sh:f, AP,$\mathchar
632\relax$1], [ Sh:208], [ Sh:638] and lately [ Sh:755]).

? DvSh:65 ?

? Sh:f ?

? Sh:208 ?

? Sh:638 ?

? Sh:755 ?

In the diamond principle for a stationary $S\subseteq\lambda$, for $\delta\in
S$ we guess $\eta\restriction\delta$ for $\eta\in{}^{\lambda}2$.

In the weak diamond, for stationary $S\subseteq\lambda$ for $\delta\in S$ we
do not guess $\eta\restriction\delta$ for $\eta\in{}^{\lambda}2$, we just
guess whether it satisfies a property - is black or white, i.e.
$\mathbb{F}(\eta\restriction\delta)\in\delimiter 696245840,1\delimiter
86405897$ for a pre-given $\mathbb{F}$. Many times it is just enough in the
construction using diamond to “avoid undesirable company”: whatever you will
do I will do otherwise. This is, of course, a considerably weaker principle,
but for $\aleph_{1}$ it follows from CH (which does not imply
$\diamondsuit_{\aleph_{1}}$) and even WCH (i.e.
$2^{\aleph_{0}}<2^{\aleph_{1}}$). In fact, it holds for many cardinals:
$2^{\lambda}>2^{<\lambda}=2^{\theta}$ suffices. Having $\mathbb{F}$ with two
colours (= values) seemed essential. In [ Sh:638], we get more colours for
$\aleph_{1}$ if the club filter on $\omega_{1}$ is saturated but this is a
heavy assumption. In [ Sh:755], if $\lambda=\lambda^{<\lambda}=2^{\mu}$, it is
proved that for “many” regular $\kappa<\lambda$, if $S\subseteq\delimiter
69624584\delta<\lambda:\text{cf}(\delta)=\kappa\delimiter 86405897$ is
stationary then we have the weak diamond for $S$. Now what does “many” mean?
For example, if $\lambda\geq\beth_{\omega}$, “many” include every large enough
regular $\kappa<\beth_{\omega}$; note that it is for “every stationary $S$”
unlike the situation, e.g. for $\aleph_{1}$ under CH (see [ Sh:f, V]). We get
there more cases if the coloring $\mathbb{F}$ on ${}^{\delta}2$ depends only
on $\eta\restriction C_{\delta}$, where
$C_{\delta}\subseteq\delta=\mathop{sup}\displaylimits(C_{\delta})$ and
$C_{\delta}$ is “small”.

? Sh:638 ?

? Sh:755 ?

? Sh:f ?

Here we get more than two colours and moreover: we guess $\eta\restriction
C_{\delta},C_{\delta}\subseteq\delta=\mathop{sup}\displaylimits(C_{\delta})$
and so $M\restriction C_{\delta}$ is $M_{\delta}$. Of course, if
$(\exists\mu<\lambda)[2^{\mu}>\lambda]$ then we cannot have
$C_{\delta}=\delta$ so this is a very reasonable restriction. (It would be
even better to have $M_{\delta}\prec M$ below but we do not know). Also for
very many $\lambda$ large enough the result holds for many regular $\kappa$.

The motivation has been trying to get better principles provably in ZFC.

We use partial squares and $I[\lambda]$, see [ Sh:108], [ AShS:221], [
Sh:237e], [ Sh:420, $\mathchar 632\relax$1] and [ DjSh:562]. We prove (in fact
this holds for quite “many” stationary $S\subseteq S^{\lambda}_{\kappa}$).

###### \stag{0.1} Theorem

For every $\mu$ strong limit of uncountable cofinality, for every regular
$\lambda>\mu$ not strongly inaccessible, for every large enough regular
$\kappa<\mu$, we have the following middle diamond: (called later MD-property)

It seems that this is stronger in sufficient measure from the weak diamond to
justify this name and we have to put the line somewhere. We shall use the
adjective middle diamond to describe even cases with colouring with more than
two values (justified by ?).

I thank Todd Eisworth for his helpful comments.

###### \stag{0.2} Theorem

In ? we can have also $\mathaccent 28694{f}=\langle f_{\delta}:\delta\in
S\rangle$ where
$f_{\delta}:u_{\delta}\rightarrow\delta,u_{\delta}\subseteq\delta,\lambda^{|u_{\delta}|}=\lambda$
such that:

then for stationarily many $\delta\in S$ we have $M_{\delta}\subseteq
M\and(f_{\delta}=f\restriction u\subseteq f)$.

###### Demonstration Proof of ? and ?

By ?(1) if $\lambda>\mu$ is regular not strongly inaccessible, then for every
large enough regular $\kappa<\mu$ the following holds:

Now without loss of generality

$\delta\in S\Rightarrow(\chi\times\theta)^{\omega}|\delta$, and let
$C^{\prime}_{\delta}=\delimiter 69624584\gamma:\gamma<\chi$ or
$(\exists\beta\in C_{\delta})(\beta\leq\gamma<\beta+\chi)$.

Lastly, let $\theta=2^{\chi+\kappa+|\tau|}$ and by ? we get the conclusion of
?.

As for ?, Clause (d) is easy by having a pairwise disjoint sequence of
suitable $\langle S_{i}:i<\lambda\rangle$ which holds by ?(2).
$\square_{?}\mskip 3.0mu\mskip 3.0mu\square_{?}$

Note that having $M_{\delta}$’s is deduced in ? from the $\theta$-Md-property
with $\theta$ large enough. Note that ? has obvious applications on various
constructions, e.g., construction of abelian groups $G$ with
Ext$(G,K)\neq\emptyset$.

Note that in ?, ?, ? instead of using “large enough
$\kappa<\mu\mathinner{\ldotp\ldotp\ldotp}\mskip 3.0mu$”, $\mu$ strong limit
and quoting [ Sh:460] we can specifically write what we use on $\kappa$ (and
on $\theta$). Also in ?, if we omit cf$(\mu)>\aleph_{0}$ (but still $\mu$ is
uncountable) the case $\lambda$ successor is not affected, and in the case
$\mu=\beth_{\delta}$ when $\omega^{2}$ divides $\delta$, the statement instead
“every large enough $\kappa$” can be replaced by “for arbitrarily large
regular $\kappa<\mu$”, or even: “for arbitrarily large strong limit
$\mu^{\prime}<\mu$, for every large enough regular $\kappa<\mu^{\prime}$”.

? Sh:460 ?

A consequence is (see more on this principle in [ Sh:667]).

? Sh:667 ?

###### Demonstration 0.1 Conclusion

1) If $\lambda,\kappa,\chi,S,\langle M_{\delta}:\delta\in S\rangle$ are as in
? and $\gamma<\kappa\vee\gamma<\chi$,

then we can find $\mathaccent 28694{\eta}=\langle\eta_{\delta}:\delta\in
S\rangle,\eta_{\delta}$ is an increasing sequence of length $\kappa$ of
ordinals $<\delta$ with limit $\delta$ such that:

then for stationarily many $\delta\in S$ we have $i<\kappa\Rightarrow
F(\eta_{\delta}(2i))=F(\eta_{\delta}(2i+1))$. 2) In fact it is enough to find
a $(\lambda,\kappa,\chi)$-Md-parameter $\langle C_{\delta}:\delta\in S\rangle$
which has the $(\gamma,2^{|\gamma|})$-Md-property (see Definition ?(5B)).

Note that if a natural quite weak conjecture on pcf holds, then we can replace
$\beth_{\omega}$ by $\aleph_{\omega}$ (and similar changes) in most places but
still need
$\lambda>\lambda_{1}>\lambda_{\delta},2^{\lambda_{1}}>2^{\lambda_{0}}\geq
2^{\aleph_{\omega}}$ in the stepping-up lemma; e.g. for every regular large
enough $\lambda$, for some $n$, we have middle diamond with $\lambda_{0}$
colours on $S^{\lambda}_{\aleph_{n}}$ \- is helpful for abelian group theory.
(Under weak conditions we get this for $\aleph_{0}$, see end of $\mathchar
632\relax$3).

Even without this conjecture it is quite hard to have the negation of this
statement as it restricts $2^{\mu}$ and cf$(2^{\mu})$ for every $\mu$!

###### Definition 0.2 Definition

1) Let “$S\subseteq\lambda$ have square” mean that ($\lambda$ is regular
uncountable, $S$ a stationary subset of $\lambda$ and): for some club $E$ of
$\lambda$ and $S^{\prime},S\cap E\subseteq S^{\prime}\subseteq\lambda$ and
there is a witness $\mathaccent 28694{C}$ which means $\mathaccent
28694{C}=\langle C_{\delta}:\delta\in
S^{\prime}\rangle,C_{\delta}\subseteq\delta$ is closed, $[\delta\in
S^{\prime}$ limit
$\Rightarrow\delta=\mathop{sup}\displaylimits(C_{\delta})],[\alpha\in
C_{\delta}\Rightarrow C_{\alpha}=\alpha\cap C_{\delta}]$ and $\delta\in
S\Rightarrow\text{ otp}(C_{\delta})<\delta$.

2) If we replace square by $(\leq\kappa)$-square this means we add
otp$(C_{\alpha})\leq\kappa$.

3) We say $(\leq\kappa,\mathchar 2\relax)$-square if above the demand
$\alpha\in C_{\delta}\Rightarrow C_{\alpha}=\alpha\cap C_{\delta}$ is
restricted to the case cf$(\alpha)\in\mathchar 2\relax$.

4) $S^{\lambda}_{\kappa}=\delimiter
69624584\delta<\lambda:\text{cf}(\delta)=\kappa\delimiter 86405897$.

Clearly ? and some results later continue Engelking Karlowic [ EK] which
continue works on density of some topological spaces, see more history in [
Sh:420].

? EK ?

? Sh:420 ?

##  $\mathchar 632\relax$1 Middle Diamond: Sufficient conditions for
cf$(2^{\mu})$

###### Definition 1.1 Definition

For regular uncountable $\lambda$,

if it is F-$(\mathchar 7\relax,\theta)$-small for some $(\lambda,\mathchar
7\relax,\theta)$-colouring $\mathbb{F}$, also called a $\theta$-colouring of
${}^{\lambda>}\mathchar 7\relax$, that is: a function $\mathbb{F}$ from
${}^{\lambda>}\mathchar 7\relax$ to $\theta$ where

w e say $S\subseteq\lambda$ is $\mathbb{F}-(\mathchar 7\relax,\theta)$-small,

if ($\mathbb{F}$ is a $\theta$-colouring of ${}^{\lambda>}\mathchar 7\relax$
and)

L et $D^{\text{md}}_{\lambda,\mathchar 7\relax,\theta}=\delimiter
69624584A\subseteq\lambda:\lambda\setminus A$ is $(\mathchar
7\relax,\theta)$-small$\delimiter 86405897$, we call it the $(\mathchar
7\relax,\theta)$-middle diamond filter on $\lambda$.

W e omit $\mathchar 7\relax$ if $\mathchar 7\relax=\theta$, writing 444this is
the weak diamond case $\theta$ instead of $(\mathchar 7\relax,\theta)$. We may
omit $\theta$ from “$S$ is $\mathbb{F}-\theta$-small”, and
$(*)^{\theta}_{\mathbb{F},S}$ when clear from the context.

If $\theta=2$ we call this weak diamond filter and may write
$D^{\text{wd}}_{\lambda,\mathchar 7\relax}$.

###### \stag{d.2} Claim

1) If $\lambda={\text{\tenrm cf\/}}(\lambda)>\aleph_{0},\lambda\geq\mathchar
7\relax\geq 2$ and $\lambda\geq\theta$

then $D^{\text{\tenrm md\/}}_{\lambda,\mathchar 7\relax,\theta}$ is a normal
filter on $\lambda$.

2) Assume $S\subseteq\lambda,2\leq\mathchar
7\relax_{\ell}\leq\lambda,2\leq\theta\leq\lambda$ for $\ell=1,2$. Then $S$ is
$(\mathchar 7\relax_{1},\theta)$-small iff $S$ is $(\mathchar
7\relax_{2},\theta)$-small 555so $\mathchar 7\relax$ seems immaterial, but
below using $\mathaccent 28694{C}$ this is not so.

3) In fact $\mathchar 7\relax_{\ell}\in[2,2^{<\lambda}]$ is enough.

###### Demonstration Proof

As in [ Sh:f, AP,$\mathchar 632\relax$1]. $\square_{?}$

Recall

###### Definition 1.2 Definition

1) Let $\chi^{<\sigma>}=\text{ Min}\delimiter
69624584|{\Cal{P}}|:{\Cal{P}}\subseteq[\chi]^{\sigma}$ and every
$A\in[\chi]^{\sigma}$ is included in the union of $<\sigma$ members of
${\Cal{P}}\delimiter 86405897$.

? Sh:f ?

2) $\chi^{<\sigma>_{\text{tr}}}=\mathop{sup}\displaylimits\delimiter
69624584|\text{lim}_{\sigma}(t)|:t$ is a tree with $\leq\chi$ nodes and
$\sigma$ levels$\delimiter 86405897$.

3) $\mathbb{U}_{J}(\mu)=\text{ Min}\delimiter
69624584|{\Cal{P}}|:{\Cal{P}}\subseteq[\mu]^{\theta}$ and for every $f:\text{
Dom}(J)\rightarrow\mu$ for some $u\in{\Cal{P}}$ we have $\delimiter
69624584i\in\text{ Dom}(J):f(i)\in u\delimiter 86405897\neq\emptyset\text{ mod
}J\delimiter 86405897$ where $J$ is an ideal on Dom$(J)$. If
$J=[\theta]^{<\theta}$ we may write $\theta$ instead of $J$.

4) Let $\chi^{[\sigma]}=\text{ Min}\delimiter
69624584|{\Cal{P}}|:{\Cal{P}}\subseteq[\chi]^{\sigma}$ and every
$A\in[\chi]^{\sigma}$ is equal to union of $<\sigma$ members of
${\Cal{P}}\delimiter 86405897$.

###### Remark 1.3 Remark

1) On $\chi^{<\sigma>_{\text{tr}}}$ see [ Sh:589], on $\chi^{<\sigma>}$ see [
Sh:430], on $\chi^{<\sigma>},\chi^{[\sigma]}$ see there and in [ Sh:460] but
no real knowledge is assumed here.

? Sh:589 ?

? Sh:430 ?

? Sh:460 ?

2) Note that when $\mathchar 7\relax=\lambda$ only
$\mathbb{F}\restriction\delimiter 69624584{}^{\delta}\delta:\delta\in
S\delimiter 86405897$ was really used. So we can restrict the colorings to
this set.

Below we may restrict ourselves to $\mathchar 7\relax=\lambda$. But if
$\mu<\lambda=\text{ cf}(\lambda),2^{\mu}=2^{\lambda}$ even for
$\theta=\mathchar 7\relax=2$, the set $\lambda$ is small (as noted by Uri
Abraham, see [ DvSh:65]).

###### Definition 1.4 Definition

1) We say $\mathaccent 28694{C}$ is a $\lambda$-Md-parameter

if:

1A) We say $\mathaccent 28694{C}$ is a $(\lambda,\kappa,\chi)$-Md-parameter if
in addition $(\forall\delta\in
S)[\text{cf}(\delta)=\kappa\wedge|C_{\delta}|<\chi]$.

? DvSh:65 ?

2) We say that F is a $(\mathaccent 28694{C},\mathchar
7\relax,\theta)$-colouring if:$\mathaccent 28694{C}=\langle
C_{\delta}:\delta\in S\rangle$ is a $\lambda$-Md-parameter and F is a function
from ${}^{\lambda>}\mathchar 7\relax$ to $\theta$ such that:

if $\delta\in S$ and $\eta_{0},\eta_{1}\in{}^{\delta}\mathchar 7\relax$ and
$\eta_{0}\restriction C_{\delta}=\eta_{1}\restriction C_{\delta}$ then
F$(\eta_{0})=$ F$(\eta_{1})$.

2A) If $\mathchar 7\relax=\theta$ we may omit it and write $(\mathaccent
28694{C},\theta)$-colouring.

2B) In part (2) we can replace F by $\mathaccent 28694{F}=\langle
F_{\delta}:\delta\in S\rangle$ where $F_{\delta}:{}^{(C_{\delta})}\mathchar
7\relax\to\theta$ such that $\eta\in{}^{\delta}\delta\wedge\delta\in
S\rightarrow\mathbb{F}(\eta)=F_{\delta}(\eta\restriction C_{\delta})$. If
$\mathchar 7\relax=\lambda$ we may use
$F_{\delta}:{}^{(C_{\delta})}\delta\rightarrow\theta$ (as we can ignore a non-
stationary $S^{\prime}\subseteq S$). So abusing notation we may write
$\mathbb{F}(\eta\restriction C_{\delta})$. Similarly if cf$(\mathchar
7\relax)=\lambda,\langle\mathchar 7\relax_{\alpha}:\alpha<\lambda\rangle$ is
increasing continuous we may restrict ourselves to
${}^{(C_{\delta})}(\mathchar 7\relax_{\delta})$.

3) Assume $\mathbb{F}$ is a $(\mathaccent 28694{C},\mathchar
7\relax,\theta)$-colouring, where $\mathaccent 28694{C}$ is a $\lambda$-Md-
parameter.

We say $\mathaccent 28694{c}\in{}^{S}\theta$ (or $\mathaccent
28694{c}\in{}^{\lambda}\theta$) is an $\mathbb{F}$-middle diamond sequence, or
an $\mathbb{F}$-Md-sequence if:

We also may say $\mathaccent 28694{c}$ is an $(\mathbb{F},S)$-Md-sequence or
an $(\mathbb{F},\mathaccent 28694{C})$-Md-sequence.

3A) We say $\mathaccent 28694{c}\in{}^{S}\theta$ is a $D-{\mathbb{F}}$-Md-
sequence if $D$ is a filter on $\lambda$ to which $S$ belongs and

4) We say $\mathaccent 28694{C}$ is a good $(\lambda,\lambda_{1})$-Md-
parameter, if $\mathaccent 28694{C}$ is a $\lambda$-Md-parameter and for every
$\alpha<\lambda$ we have $\lambda_{1}>|\delimiter
69624584C_{\delta}\cap\alpha:\delta\in S\and\alpha\in C_{\delta}\delimiter
86405897|$. If $\lambda_{1}=\lambda$ we may write “good $\lambda$-Md-
parameter”. Similarly for $(\lambda,\lambda_{1},\kappa,\chi)$-good-Md-
parameter and $(\lambda,\kappa,\chi)$-good-Md-parameter (i.e. adding
$\mathaccent 28694{C}$ is a $(\lambda,\kappa,\chi)$-Md-parameter.

4A) We say $\mathaccent 28694{C}$ is weakly good $\lambda$-Md-parameter if it
is a $\lambda$-Md-parameter and for every $\alpha<\lambda$ we have
$\lambda>|\delimiter 69624584C_{\delta}\cap\alpha:\delta\in
S\and\alpha\in\text{ nacc}(C_{\delta})\delimiter 86405897|$. Similarly a
weakly good $(\lambda,\kappa,\chi)$-Md-parameter.

4B) We say $\mathaccent 28694{C}$ is good${}^{+}-\lambda$-Md-parameter if
$\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$ is a
$\lambda$-Md-parameter and $\alpha\in C_{\delta_{1}}\cap
C_{\delta_{2}}\Rightarrow C_{\delta_{1}}\cap\alpha=C_{\delta_{2}}\cap\alpha$.
Similarly good${}^{+}-(\lambda,\kappa,\chi)$-Md-parameter.

4C) We define weakly good${}^{+}-\lambda$-Md-parameter or
$(\lambda,\kappa,\chi)$-Md-parameter parallely, i.e., $\alpha\in\text{
nacc}(C_{\delta_{1}})\cap\text{ nacc}(C_{\delta_{2}})\Rightarrow
C_{\delta_{1}}\cap\delta_{1}=C_{\delta_{2}}\cap\delta_{1}$.

###### Definition 1.5 Definition

1) We say $\mathaccent 28694{C}$ has the $(D,\mathchar 7\relax,\theta)$-Md-
property

if

then there is a $D-\mathbb{F}$-Md-sequence, see Definition 1.4, part (3A). 2)
We say that $\mathaccent 28694{C}$ has the $(D,\theta)$-Md-property as in part
(1) only $\mathbb{F}$ is a $(\mathaccent 28694{C},\theta,\theta)$-colouring.
We may omit $D$ if $D$ is the club filter. We may omit $\theta$ if $\theta=2$.

3) We say $S\subseteq\lambda$ has the $(\mathchar 7\relax,\theta)$-Md-property
mean that some $\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$
has, and we say the $\theta$-Md-property if $\mathchar 7\relax=\theta$.

###### Remark 1.6 Remark

We may consider replacing $C_{\delta}$ by $(C_{\delta},D_{\delta}),D_{\delta}$
a filter on $C_{\delta}$, that is $F_{\delta}(\eta)$ depend on
$(\eta\restriction C_{\delta})/D_{\delta}$ only. For the time being it does
not make a real difference.

###### \stag{d.5a} Claim

1) The restrictions in 1.4(2B) are O.K.

2) In Definition 1.4(4), if $\lambda_{1}>\lambda$

then goodness holds trivially.

3) If $\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$ is a weakly
good $(\lambda,\kappa,\chi)$-parameter, then for some club $C$ of $\lambda$
there is $\mathaccent 28694{C}^{\prime}=\langle C^{\prime}_{\delta}:\delta\in
S\cap E\rangle$ which is a good${}^{+}\mskip
3.0mu(\lambda,\kappa,\chi)$-parameter.

###### Demonstration Proof

Easy. (Note that the $C^{\prime}_{\delta}$’s is not required to be a club of
$\delta$). Concerning part (3), see [ Sh:420, $\mathchar 632\relax$1]).
$\square_{?}$

? Sh:420 ?

###### \stag{d.6} The Main Claim

1) Assume

Then $\mathaccent 28694{C}$ has the $(D,\lambda,\theta)$-Md-property (recall
that this means that the number of colours is $\theta$ not just 2).

2) Moreover, $\mathaccent 28694{C}$ has the $(D,2^{\mu},\theta)$-Md-property.

We may wonder if clause (f) of the assumption is reasonable; the following
Claim gives some sufficient conditions.

###### \stag{d.7} Claim

Clause (f) of ? holds, i.e., Sep$(\mu,\theta)$ holds,

if at least one of the following holds:

We shall prove this later.

###### Remark 1.7 Remark

1) Note that $\lambda=\lambda^{<\lambda}=\chi,\kappa=\text{
cf}(\kappa)<\lambda,S=S^{\lambda}_{\kappa},C_{\delta}=\delta$ is a reasonable
case and that $\langle C_{\delta}:\delta\in S^{\lambda}_{\kappa}\rangle$ is a
$(\lambda,\kappa,\lambda)$-good Md-parameter.

2) We can replace $\kappa$ by $*$ meaning $[\delta\in S\Rightarrow\delta$ is a
regular cardinal]. Then “$|\alpha|^{<\kappa>_{\text{tr}}}<2^{\mu}$” is
replaced by “$\delta\in S\Rightarrow\delta^{<\delta>_{\text{tr}}}<2^{\mu}$”.

3) We may consider finding a guessing sequence $\mathaccent 28694{c}$
simultaneously for many colourings. In the notation of [ Sh:f, AP,$\mathchar
632\relax$1] this is the same as having large $\mu(0)$. We may reconsider the
problem of $\lambda^{+}$-entangled linear ordering where
$\lambda=\lambda^{\aleph_{0}}$, see [ Sh:462].

? Sh:f ?

? Sh:462 ?

4) For $\theta=2$, (i.e. weak diamond) the proof below can be simplified: we
can forget ${\Cal{F}}$ and choose $\varrho_{\delta}\in{}^{\mu}\theta\delimiter
2548495\delimiter 69624584\rho_{\eta}:\eta\in{\Cal{P}}_{\delta}\delimiter
86405897$ such that for some $i<\mu$ the sequence
$\langle\varrho_{\delta}(i):\delta\in S\rangle$ is a weak diamond sequence for
$\mathbb{F}$.

5) Note that the “minimal” natural choices for getting $\theta$-MD property
are $\kappa=\aleph_{0},\theta=2^{\aleph_{0}},\mu=2^{\theta},\lambda=\text{
cf}(2^{\mu})$.

###### Demonstration Proof of ?

Let $\mathbb{F}$ be a given $(\mathaccent 28694{C},2^{\mu},\theta)$-colouring.

Let ${\Cal{F}}=\delimiter 69624584f_{\xi}:\xi<\mu\delimiter 86405897$ where
${\Cal{F}}$ is from clause (f) of our assumptions, and for
$\rho\in{}^{\mu}\theta$ let Sol${}_{\rho}=:\delimiter
69624584\nu\in{}^{\mu}\theta:\text{ for every }\varepsilon<\mu$ we have
$\rho(\varepsilon)\neq f_{\varepsilon}(\nu)\delimiter 86405897$ where Sol
stands for solutions, so by our assumption $(f)(\beta)$ it follows that
$\rho\in{}^{\mu}\theta\Rightarrow|\text{Sol}_{\rho}|<\mathchar 7\relax$.

Let $h$ be an increasing continuous function from $\lambda$ into $2^{\mu}$
with unbounded range. Let cd be a one-to-one function from ${}^{\mu}(2^{\mu})$
onto $2^{\mu}$ such that, for simplicity, and without loss of generality

we have

$\alpha=\text{
cd}(\langle\alpha_{\varepsilon}:\varepsilon<\mu\rangle)\Rightarrow\alpha\geq\mathop{sup}\displaylimits\delimiter
69624584\alpha_{\varepsilon}:\varepsilon<\mu\delimiter 86405897$

and let the function cd${}_{i}:2^{\mu}\rightarrow 2^{\mu}$ for $i<\mu$ be such
that
cd${}_{i}(\langle\text{cd}(\alpha_{\varepsilon}:\varepsilon<\mu)\rangle)=\alpha_{i}$;
recall $\lambda=\text{ cf}(2^{\mu})\geq\chi$.

Let ${\Cal{C}}=\delimiter 69624584C_{\delta}\cap\alpha:\delta\in S\text{ and
}\alpha\in C_{\delta}\delimiter 86405897$, recall that
$|{\Cal{C}}|\leq\lambda\leq 2^{\mu}$ and let $T=\delimiter 69624584\eta:$ for
some $C\in{\Cal{C}}$ (hence $C$ is of cardinality $<\chi$), we have
$\eta\in{}^{C}(2^{\mu})\delimiter 86405897$, so by assumptions (c) + (d)
clearly $|T|=\displaystyle\mathchar
4944\relax\displaylimits_{C\in{\Cal{C}}}2^{\mu\mathchar
8705\relax|C|}\leq|{\Cal{C}}|\times(2^{\mu})^{<\chi}\leq 2^{\mu}$ but $|T|\geq
2^{\mu}$ hence $|T|=2^{\mu}$, so let us list $T$ with no repetitions as
$\delimiter 69624584\eta_{\alpha}:\alpha<2^{\mu}\delimiter 86405897$ and let
$T_{<\alpha}=\delimiter 69624584\eta_{\beta}:\beta<h(\alpha)\delimiter
86405897$ for $\alpha<\lambda$. For $\delta\in S$ let
${\Cal{P}}_{\delta}=:\delimiter 69624584\eta:\eta$ is a function from
$C_{\delta}$ into $h(\delta)$ such that for every $\alpha\in C_{\delta}$ we
have $\eta\restriction(C_{\delta}\cap\alpha)\in T_{<\delta}\delimiter
86405897$.

As $\mathaccent 28694{C}$ is a $(\lambda,\kappa,\chi)$-Md-parameter we have
$\delta\in S\Rightarrow\text{ cf}(\delta)=\kappa$. So by clause (e) of our
assumptions necessarily ${\Cal{P}}_{\delta}$ has cardinality $<2^{\mu}$. For
each $\eta\in{\Cal{P}}_{\delta}$ and $\varepsilon<\mu$ we define
$\nu_{\eta,\varepsilon}\in{}^{C_{\delta}}(h(\delta))$ by
$\nu_{\eta,\varepsilon}(\alpha)=\text{ cd}_{\varepsilon}(\eta(\alpha))$ for
$\alpha\in C_{\delta}$. Now for $\eta\in{\Cal{P}}_{\delta}$, clearly
$\rho_{\eta}=:\langle\mathbb{F}(\nu_{\eta,\varepsilon}):\varepsilon<\mu\rangle$
belongs to ${}^{\mu}\theta$; see end of ?(2B). Clearly $\delimiter
69624584\rho_{\eta}:\eta\in{\Cal{P}}_{\delta}\delimiter 86405897$ is a subset
of ${}^{\mu}\theta$ of cardinality $\leq|{\Cal{P}}_{\delta}|$ which as said
above is $<2^{\mu}$. Recall the definition of Solρ from the beginning of the
proof and remember clause (f)$(\beta)$ of the assumption of our claim ?, so
$\eta\in{\Cal{P}}_{\delta}\Rightarrow\text{ Sol}_{\rho_{\eta}}$ has
cardinality $<\mathchar 7\relax=\text{ cf}(\mathchar 7\relax)\leq 2^{\mu}$, so
we can find $\varrho^{*}_{\delta}\in{}^{\mu}\theta\setminus\cup\delimiter
69624584\text{Sol}_{\rho_{\eta}}:\eta\in{\Cal{P}}_{\delta}\delimiter
86405897$. Clearly $\delimiter
69624584\eta\restriction(C_{\delta}\cap\alpha):\alpha\in C_{\delta}\delimiter
86405897$ is naturally a tree with ${\Cal{P}}_{\delta}$ the set of
$\kappa$-branches and set of nodes $\subseteq T_{<\delta}$ hence with
$<\lambda$ nodes.

For each $\varepsilon<\mu$ recall that $\varrho^{*}_{\delta}\in{}^{\mu}\theta$
for $\delta\in S$ and $f_{\varepsilon}$ from the list of ${\Cal{F}}$ is a
function from ${}^{\mu}\theta$ to $\theta$ so
$f_{\varepsilon}(\varrho^{*}_{\delta})<\theta$ hence we can consider the
sequence $\mathaccent 28694{c}^{\varepsilon}=\langle
f_{\varepsilon}(\varrho^{*}_{\delta}):\delta\in S\rangle\in{}^{S}\theta$ as a
candidate for being an F-Md-sequence. If one of them is, we are done. So
assume toward contradiction that for each $\varepsilon<\mu$ there is a
sequence $\eta_{\varepsilon}\in{}^{\lambda}(2^{\mu})$ that exemplifies its
failure, so there is a $E_{\varepsilon}\in D$ such that

Define $\eta^{*}\in{}^{\lambda}(2^{\mu})$ by $\eta^{*}(\alpha)=\text{
cd}(\langle\eta_{\varepsilon}(\alpha):\varepsilon<\mu\rangle)$, now as
$\lambda$ is regular uncountable and the choice of $h$ clearly $E=:\delimiter
69624584\delta<\lambda:$ for every $\alpha<\delta$ we have
$\eta^{*}(\alpha)<h(\delta)$ and if $\delta^{\prime}\in S,\alpha\in
C_{\delta^{\prime}}\cap\delta$ and $C^{\prime}=C_{\delta^{\prime}}\cap\alpha$
then $\eta^{*}\restriction C^{\prime}\in T_{<\delta}\delimiter 86405897$ is a
club of $\lambda$ (see the choice of $T$ and $T_{<\delta}$, and recall as well
that by assumption (c) the sequence $\mathaccent 28694{C}$ is good or just
weakly good, see Definition 1.4(4) + (4A)).

By clause (a) in the assumption of our claim ?, the filter $D$ includes the
clubs of $\lambda$ so clearly $E\in D$; also $D$ is $\mu^{+}$-complete and
every club of $\lambda$ belongs to it by assumption (a) hence
$E^{*}=:\cap\delimiter 69624584E_{\varepsilon}:\varepsilon<\mu\delimiter
86405897\cap E$ belongs to $D$. Now choose $\delta\in E^{*}\cap S$, fixed for
the rest of the proof; clearly $\eta^{*}\restriction
C_{\delta}\in{\Cal{P}}_{\delta}$; just check the definitions of
${\Cal{P}}_{\delta}$ and $E,E^{*}$. Let $\varepsilon<\mu$. Now recall that
$\nu_{\eta^{*}\restriction C_{\delta},\varepsilon}$ is the function from
$C_{\delta}$ to $h(\delta)$ defined by

$\nu_{\eta^{*}\restriction C_{\delta},\varepsilon}(\alpha)=\text{
cd}_{\varepsilon}(\eta^{*}(\alpha)).$

But by our choice of $\eta^{*}$ clearly $\alpha\in C_{\delta}\Rightarrow\text{
cd}_{\varepsilon}(\eta^{*}(\alpha))=\eta_{\varepsilon}(\alpha)$, so

$\alpha\in C_{\delta}\Rightarrow\nu_{\eta^{*}\restriction
C_{\delta},\varepsilon}(\alpha)=\eta_{\varepsilon}(\alpha)\quad\text{ so
}\quad\nu_{\eta^{*}\restriction
C_{\delta},\varepsilon}=\eta_{\varepsilon}\restriction C_{\delta}.$

Hence $\mathbb{F}(\nu_{\eta^{*}\restriction
C_{\delta},\varepsilon})=\mathbb{F}(\eta_{\varepsilon}\restriction
C_{\delta})$, however as $\delta\in E^{*}\subseteq E_{\varepsilon}$ clearly
$\mathbb{F}(\eta_{\varepsilon}\restriction C_{\delta})\neq
f_{\varepsilon}(\varrho^{*}_{\delta})$ and as $\eta^{*}\restriction
C_{\delta}\in{\Cal{P}}_{\delta}$ clearly $\rho_{\eta^{*}\restriction
C_{\delta}}\in{}^{\mu}\theta$ is well defined. Now easily
$\rho_{\eta^{*}\restriction
C_{\delta}}(\varepsilon)=\mathbb{F}(\nu_{\eta^{*}\restriction
C_{\delta},\varepsilon})$ by the definition of $\rho_{\eta^{*}\restriction
C_{\delta}}$, together $\rho_{\eta^{*}\restriction
C_{\delta}}(\varepsilon)\neq f_{\varepsilon}(\varrho^{*}_{\delta})$.

As this holds for every $\varepsilon<\mu$ it follows by the definition of
Sol${}_{\rho_{\eta^{*}\restriction C_{\delta}}}$ that
$\varrho^{*}_{\delta}\in\text{Sol}_{\rho_{\eta^{*}\restriction C_{\delta}}}$.
But $\eta^{*}\restriction C_{\delta}\in{\Cal{P}}_{\delta}$ was proved above so
$\rho_{\eta^{*}\restriction C_{\delta}}\in\delimiter
69624584\rho_{\eta}:\eta\in{\Cal{P}}_{\delta}\delimiter 86405897$ hence
$\varrho^{*}_{\delta}\in\text{ Sol}_{\rho_{\eta^{*}\restriction
C_{\delta}}}\subseteq\cup\delimiter
69624584\text{Sol}_{\rho_{\nu}}:\nu\in{\Cal{P}}_{\delta}\delimiter 86405897$
whereas $\varrho^{*}_{\delta}$ has been chosen outside this set,
contradiction. $\square_{?}$

###### Definition 1.8 Definition

1) Sep$(\chi,\mu,\theta,\theta_{1},\mathchar 7\relax)$ means that for some
$\mathaccent 28694{f}$:

2) We may omit $\theta_{1}$ if $\theta=\theta_{1}$. We write Sep$(\mu,\theta)$
if for some $\mathchar 7\relax=\text{ cf}(\mathchar 7\relax)\leq 2^{\mu}$ we
have Sep$(\mu,\mu,\theta,\theta,\mathchar 7\relax)$.

There are obvious monotonicity properties:

###### \stag{d.8z} Claim

If Sep$(\chi,\mu,\theta,\theta_{1},\mathchar 7\relax)$ holds and
$\chi^{\prime}\geq\chi,\mu^{\prime}=\mu,\theta^{\prime}\leq\theta,\theta^{\prime}_{1}\leq\theta_{1}$
and $\mathchar 7\relax^{\prime}\geq\mathchar 7\relax$,

then
Sep$(\chi^{\prime},\mu^{\prime},\theta^{\prime},\theta^{\prime}_{1},\mathchar
7\relax^{\prime})$ holds.

###### Demonstration Proof

Let $\mathaccent 28694{f}=\langle f_{\varepsilon}:\varepsilon<\chi\rangle$
exemplify Sep$(\chi,\mu,\theta,\theta_{1},\mathchar 7\relax)$. For
$\varepsilon<\chi^{\prime}$ we define
$f^{\prime}_{\varepsilon}:{}^{(\mu^{\prime})}(\theta^{\prime})\rightarrow\theta^{\prime}_{1}$
by: $f^{\prime}_{\varepsilon}(\nu)=f_{\varepsilon}(\nu)$ if
$\varepsilon<\chi\and f_{\varepsilon}(\nu)<\theta^{\prime}_{1}$ and
$f^{\prime}_{\varepsilon}(\nu)=0$ otherwise; note that
${}^{(\mu^{\prime})}(\theta^{\prime})\subseteq{}^{\mu}\theta$ hence
$f_{\varepsilon}(\nu)$ is well defined if $\varepsilon<\chi$. Let
$\varrho^{\prime}\in{}^{(\chi^{\prime})}(\theta^{\prime}_{1})$ and
$A^{\prime}_{\varrho^{\prime}}=\delimiter
69624584\nu\in{}^{(\mu^{\prime})}(\theta^{\prime})$: for every
$\varepsilon<\chi^{\prime}$ we have
$f^{\prime}_{\varepsilon}(\nu)\neq\varrho^{\prime}(\varepsilon)\delimiter
86405897$. Let $\varrho=:\varrho^{\prime}\restriction\chi$ so
$\varrho\in{}^{\chi}(\theta^{\prime}_{1})\subseteq{}^{\chi}(\theta_{1})$,
hence by the choice of $\langle f_{\varepsilon}:\varepsilon<\chi\rangle$ the
set $A_{\varrho}=\delimiter 69624584\nu\in{}^{\mu}\theta$: for every
$\varepsilon<\chi$ we have
$f_{\varepsilon}(\nu)\neq\varrho(\varepsilon)\delimiter 86405897$ has
cardinality $<\mathchar 7\relax$. Recall that $\mu^{\prime}=\mu$, and assume
$\nu\in A^{\prime}_{\varrho^{\prime}}$ then
$\nu\in{}^{\mu}(\theta^{\prime})\subseteq{}^{\mu}\theta$, and for
$\varepsilon<\chi^{\prime}$ we have
$f^{\prime}_{\varepsilon}(\nu)\neq\varrho^{\prime}(\varepsilon)$ hence for
$\varepsilon<\chi(\leq\chi^{\prime})$ we have
$f_{\varepsilon}(\nu)<\theta^{\prime}\and\varrho(\varepsilon)=\varrho^{\prime}(\varepsilon)\neq
f^{\prime}_{\varepsilon}(\nu)=f_{\varepsilon}(\nu)$ or
$f_{\varepsilon}(\nu)\geq\theta^{\prime}\and\varrho(\varepsilon)=\varrho^{\prime}(\varepsilon)<\theta^{\prime}\and
f_{\varepsilon}(\nu)\neq\varrho(\varepsilon)$ so
$f_{\varepsilon}(\nu)\neq\varrho(\varepsilon)$. Hence $\nu\in
A^{\prime}_{\varrho^{\prime}}\Rightarrow\nu\in A_{\varrho}$ and hence
$|A^{\prime}_{\varrho^{\prime}}|\leq|A_{\varrho}|<\mathchar 7\relax$ and we
are done. $\square_{?}$

To show that ? has reasonable assumptions we should still prove ?.

###### Demonstration Proof of ?

Our claim gives sufficient conditions for Sep$(\mu,\theta)$, i.e.
Sep$(\mu,\mu,\theta,\theta,\mathchar 7\relax)$ for some $\mathchar
7\relax=\text{ cf}(\mathchar 7\relax)\leq 2^{\mu}$. Clearly if $\mathchar
7\relax<2^{\mu}$ we can waive “$\mathchar 7\relax$ regular” as $\mathchar
7\relax^{+}$ can serve as well. Let $\chi=\mu$.

Case 1: $\mu=\mu^{\theta},\mathchar
7\relax=\theta,\theta_{0}\in[\theta,\mu],\theta_{1}=\theta$ and we shall prove
Sep$(\mu,\mu,\theta_{0},\theta_{1},\mathchar 7\relax)$. Let

$\displaystyle{\Cal{F}}=\biggl{\delimiter 69624584}f:$ $\displaystyle f\text{
is a function with domain }{}^{\mu}(\theta_{0})\text{ we have}$
$\displaystyle\text{for some }u\in[\mu]^{\theta}\text{ and }\mathaccent
28694{\rho}=\langle\rho_{i}:i<\theta\rangle$ $\displaystyle\text{ with no
repetition},\rho_{i}\in{}^{u}(\theta_{0})\text{ we have}$
$\displaystyle(\forall\nu\in{}^{\mu}(\theta_{0})[\rho_{i}\subseteq\nu\Rightarrow
f(\nu)=i]\text{ and}$
$\displaystyle(\forall\nu\in{}^{\mu}(\theta_{0})[(\mathchar
4950\relax\displaylimits_{i<\theta}\rho_{i}\nsubseteq\nu)\Rightarrow
f(\nu)=0]\biggr{\delimiter 86405897}.$

We write $f=f^{*}_{u,\mathaccent 28694{\rho}}$, if $u,\mathaccent 28694{\rho}$
witness $f\in{\Cal{F}}$ as above.

Recalling $\mu=\mu^{\theta}$, clearly $|{\Cal{F}}|=\mu$. Let
${\Cal{F}}=\delimiter 69624584f_{\varepsilon}:\varepsilon<\mu\delimiter
86405897$ and we let $\mathaccent 28694{f}=\langle
f_{\varepsilon}:\varepsilon<\mu\rangle$ of course. Clearly clauses (a),(b) of
Definition 1.8 holds and let us check clause (c). So suppose
$\varrho\in{}^{\mu}\theta$ and let $R=R_{\varrho}=:\delimiter
69624584\nu\in{}^{\mu}(\theta_{0})$: for every $\varepsilon<\mu$ we have
$f_{\varepsilon}(\nu)\neq\varrho(\varepsilon)\delimiter 86405897$, we have to
prove $|R|<\theta$ as $\mathchar 7\relax=\theta$.

Towards contradiction, assume that $R\subseteq{}^{\mu}(\theta_{0})$ has
cardinality $\geq\theta$ choose $R^{\prime}\subseteq R$ of cardinality
$\theta$ and so we can find $u\in[\mu]^{\theta}$ such that
$\langle\nu\restriction u:\nu\in R^{\prime}\rangle$ is without repetitions.

Let $\delimiter 69624584\nu_{i}:i<\theta\delimiter 86405897$ list $R^{\prime}$
without repetitions and let $\rho_{i}=:\nu_{i}\restriction u$. Now let
$\mathaccent 28694{\rho}=\langle\rho_{i}:i<\theta\rangle$, so
$f^{*}_{u,\mathaccent 28694{\rho}}$ is well defined and belong to ${\Cal{F}}$
hence for some $\zeta<\mu$ we have $f^{*}_{u,\mathaccent
28694{\rho}}=f_{\zeta}$. Now for each $i<\theta,\nu_{i}\in R^{\prime}\subseteq
R$ hence by the definition of
$R,(\forall\varepsilon<\mu)(f_{\varepsilon}(\nu_{i})\neq\varrho(\varepsilon))$
in particular for $\varepsilon=\zeta$ we get
$f_{\zeta}(\nu_{i})\neq\varrho(\zeta)$. But by the choice of
$\zeta,f_{\zeta}(\nu_{i})=f^{*}_{u,\mathaccent 28694{\rho}}(\nu_{i})$ and by
the definition of $f^{*}_{u,\mathaccent 28694{\rho}}$ recalling
$\nu_{i}\restriction u=\rho_{i}$ we have $f^{*}_{u,\mathaccent
28694{\rho}}(\nu_{i})=i$, so $i=f_{\zeta}(\nu_{i})\neq\varrho(\zeta)$. This
holds for every $i<\theta$ whereas
$\varrho\in{}^{\mu}(\theta_{1})={}^{\mu}\theta$, contradiction. Case 2:
$2^{\theta}\leq\mu$ and $\mathbb{U}_{\theta}[\mu]=\mu$ and $\mathchar
7\relax=(2^{\theta})^{+}$ or just $(2^{<\theta})^{+}$ hence $\mathchar
7\relax=\text{ cf}(\mathchar 7\relax)\leq\mu^{+}\leq 2^{\mu}$.

Let $\delimiter 69624584u_{i}:i<\mu\delimiter 86405897\subseteq[\mu]^{\theta}$
exemplify $\mathbb{U}_{\theta}[\mu]=\mu$. Define ${\Cal{F}}$ as in case 1
except that for notational simplicity $\theta_{0}=\theta_{1}=\theta$ and we
restrict ourselves to $u\in\cup\delimiter
69624584{\Cal{P}}(u_{i}):i<\mu\delimiter 86405897$. Clearly
$|{\Cal{F}}|\leq\mu$.

Assume that $R\subseteq{}^{\mu}\theta$ and $|R|\geq(2^{\theta})^{+}$, hence we
can find $\rho^{*},\langle(\alpha_{\zeta},\rho_{\zeta}):i<\theta\rangle$ such
that:

$\rho^{*},\rho_{\zeta}\in R$

$\rho_{\zeta}\restriction\delimiter 69624584\alpha_{\xi}:\xi<\zeta\delimiter
86405897=\rho^{*}\restriction\delimiter
69624584\alpha_{\xi}:\xi<\zeta\delimiter 86405897$

$\rho_{\zeta}(\alpha_{\zeta})\neq\rho^{*}(\alpha_{\zeta}).$

Clearly $\langle\alpha_{\zeta}:\zeta<\theta\rangle$ is with no repetitions.

So for some $i<\mu$ the set $u_{i}\cap\delimiter
69624584\alpha_{\zeta}:\zeta<\theta\delimiter 86405897$ has cardinality
$\theta$, call it $u$. So $\delimiter 69624584\rho\restriction u:\rho\in
R\delimiter 86405897$ has cardinality $\geq\theta$ and we can continue as in
Case 1. If we like to use $\theta_{0}\in[\theta,\mu]$ let pr be a pairing
function on $\mu$, without loss of generalityeach $u_{i}$ is closed under pr
and its inverses, and $f_{u,\mathaccent 28694{\rho}}$ is defined iff for some
$\varepsilon<\mu$ we have $u\subseteq u_{\varepsilon}$ and
$i<\theta\Rightarrow\text{ Rang}(\rho_{i})\subseteq u_{\varepsilon}$. In the
end choose $i$ such that $u_{i}\cap\delimiter 69624584{\text{\tenrm
pr\/}}(\alpha_{\zeta},{\text{\tenrm
pr\/}}(\rho_{\zeta}(\alpha_{\zeta}),\rho^{*}(\alpha_{\zeta})):\zeta<\theta\delimiter
86405897$ has cardinality $\theta$.

Case 3: $\mathbb{U}_{J}[\mu]=\mu$ where $J=[\sigma]^{<\theta}$ and
$2^{\sigma}\leq\mu$ and $\mathchar 7\relax=[2^{<\sigma}]^{+}$.

The proof is similar to Case 2, only find now $\rho^{*}$ and
$\langle(\alpha_{\zeta},\rho_{\zeta}):\zeta<\sigma\rangle$ and choose $i<\mu$
such that $u_{i}\cap\delimiter
69624584\text{pr}(\alpha_{\zeta},\rho_{\zeta}(\alpha_{\zeta})):\zeta<\sigma\delimiter
86405897$ has cardinality $\theta$.

Case 4: $\mu>\theta\neq\text{ cf}(\mu)$ and $\mu$ is strong limit.

Follows by case 2.

Case 5: $\mu\geq\beth_{\omega}(\theta)$.

By [ Sh:460] we can find regular $\sigma<\beth_{\omega}(\theta)$ which is
$>\theta$ and such that $\mathbb{U}_{\sigma}[\mu]=\mu$, so Case 3 applies.

? Sh:460 ?

Note that in case 2, we can replace $2^{\theta}\leq\mu$ by: there is
${\Cal{F}}\subseteq{}^{\theta}\theta$ of cardinality $\leq\mu$ such that for
every $A\in[\theta]^{\theta}$ for some $f\in{\Cal{F}}$ we have $\theta=\text{
Rang}(f\restriction A)$. The proof is obvious. Parallely in case 3.
$\square_{?}$

There are obvious monotonicity properties.

###### \stag{d.11} Claim

1) If $(\lambda,\mathaccent 28694{C})$ has the $\theta$-Md-property (see
Definition 1.5(1)) and $\mathaccent 28694{C}=\langle C_{\delta}:\delta\in
S\rangle$ and $C^{1}_{\delta}$ is an unbounded subset of $C_{\delta}$ for
$\delta\in S$

then $(\lambda,\langle C^{1}_{\delta}:\delta\in S\rangle)$ has a $\theta$-Md
property. We can also increase $S$ and decrease $\theta$.

2) Assume $\lambda={\text{\tenrm
cf\/}}(\lambda)>\aleph_{0},S\subseteq\delimiter
69624584\delta:\delta<\lambda,{\text{\tenrm cf\/}}(\delta)=\kappa\delimiter
86405897$ is stationary and $\kappa<\chi\leq\lambda$.

Then some $\langle C_{\delta}:\delta\in S\rangle$ is a
$(\lambda,\kappa,\chi)$-Md-parameter.

3) Assume $\lambda=\text{ cf}(\lambda)>\kappa=\text{ cf}(\kappa),S\subseteq
S^{\lambda}_{\kappa}$ and

Then $\mathaccent 28694{C}^{2}$ has the $(\mathchar 7\relax,\theta)$-Md-
property.

4) If $\mathaccent 28694{C}$ is a $(\lambda,\kappa,\chi^{+})$-Md-parameter
with the $(\mathchar 7\relax,\theta)$-Md-property, $\mathchar
7\relax^{\chi}=\mathchar 7\relax$ and $C^{\prime}_{\delta}=\delimiter
69624584\alpha+i:\alpha\in C_{\delta}\vee\alpha=0$ and $i<\chi\delimiter
86405897$, then $\mathaccent 28694{C}^{\prime}=\langle
C^{\prime}_{\delta}:\delta\in{\text{\tenrm Dom\/}}(\mathaccent
28694{C})\rangle$ is a $(\lambda,\kappa,\chi^{+})$-Md-parameter with the
$\theta$-Md-property.

###### Demonstration Proof

1) Obvious.

2) Easy (note: goodness is not required).

3), 4) Easy by coding (as in the proof of ? below). $\square_{?}$

We may consider the following generalization.

###### Definition 1.9 Definition

1) We say $\mathaccent 28694{C}$ is a $\lambda$-MD1-parameter

if:

2) We say that a $\lambda$-MD1-parameter $\mathaccent 28694{C}$ is good if
$\alpha<\lambda\Rightarrow\lambda>|\delimiter 69624584\displaystyle\mathchar
4947\relax\displaylimits_{\varepsilon<\zeta}C_{\delta,\varepsilon}:\delta\in
S,\zeta<\text{ cf}(\delta)$ and
$\alpha\geq\mathop{sup}\displaylimits(\displaystyle\mathchar
4947\relax\displaylimits_{\varepsilon<\zeta}C_{\delta,\varepsilon})\delimiter
86405897|$.

3) The $(D,\mathchar 7\relax,\theta)$-Md1-property for $\mathaccent 28694{C}$
means for $\langle C_{\delta}:\delta\in S\rangle$, similarly for the other
variants of this property (so the $(D,\mathchar 7\relax,\theta)$-MD1-property
for $\lambda$ for the stationary $S\subseteq S^{\lambda}_{\kappa}$ is the same
as the $(D,\mathchar 7\relax,\theta)$-Md-property).

We may consider the following variant (and claims on it)

###### \stag{d.11b} Claim

Assume

Then $\langle C_{\delta},C^{*}_{\delta,\varepsilon}:\delta\in
S^{*},\varepsilon<\text{ cf}(\delta)\rangle$ is a good
$(S,\kappa,\chi)$-MD1-parameter.

###### Demonstration Proof

Easy.

###### \stag{d.11c} Claim

1) Assume $\mathaccent 28694{C}=\langle
C_{\delta},C_{\delta,\varepsilon}:\delta\in S,\varepsilon<\text{
cf}(\delta)\rangle$ is a good $(\lambda,\kappa,\chi)$-MD0-parameter.

Then claim ? holds (i.e. we get the $(D,\theta)$-MD1-property, of course, we
naturally change assumption (e) of ?).

2) In part (1) hence in ?, we can weaken clause (e) to

###### Demonstration Proof

Same proof as ?.

The following claim shows the close affinity of the properties stated in the
theorems in the introduction and the $\theta$-Md-property with which we deal
here, also it is needed to deduce them from ?.

###### \stag{d.12} Claim

1) Assume $(\lambda,\mathaccent 28694{C})$ has the $\theta$-Md-property where
$\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$ is a
$(\lambda,\kappa,\chi)$-Md-parameter for $D$. If $\theta=\theta^{<\chi}$

then $(\lambda,\mathaccent 28694{C})$ has the $(\chi,\theta)$-MD0-property;
which means that we can find $\langle f_{\delta}:\delta\in S\rangle$ such
that:

If we omit $\chi$ we mean $\chi=2$ so $\alpha^{*}_{\delta}=1$ for every
$\delta\in S$.

2) Assume in addition that $\mathaccent 28694{C}$ is good${}^{+}\mskip
3.0mu(\lambda,\kappa,\chi)$-MD-parameter then $(\lambda,\mathaccent 28694{C})$
has the $(\chi,\theta)$-MD2-property which means: if $\tau$ is a relational
vocabulary of cardinality $<\chi$, then we can find $\langle
M_{\delta}:\delta\in S\rangle$ such that:

then the set $A_{M,\alpha}=\delimiter 69624584\delta\in S:M\restriction
C_{\delta}=M_{\delta}$ and $\alpha^{*}_{\delta}=\alpha\delimiter 86405897$ is
stationary and even belongs to $D^{+}$. If we omit $\chi$ we mean $\chi=2$ so
$\alpha^{*}_{\delta}=1$ for every $\delta\in S$.

3) In part (2) we can allow $\tau$ to have function symbols but have:

then the following set belongs to $D^{+}$

$A^{\prime}_{M,\alpha}=\delimiter 69624584\delta\in
S:\alpha^{*}_{\delta}=\alpha\text{ and }(M\restriction
c\ell_{M}(C^{\prime}_{\delta}),c)_{c\in
C^{\prime}_{\delta}}\cong(M_{\delta},c)_{c\in C^{\prime}_{\delta}}\delimiter
86405897.$

4) In part (2) we can replace good+ by good if we add

###### Demonstration Proof

This is a translation (note: “$\tau$ is a relational” is a strong
restriction).

1) First fix $\alpha^{*}<\chi$. Let cd be a one-to-one function from
$\theta^{<\chi}$ onto $\theta$. Define $\langle F_{\delta}:\delta\in S\rangle$
as follows: if $f\in{}^{(C_{\delta})}\theta$ then $F_{\delta}(f)=\text{
cd}(\langle f(\beta_{\delta,\varepsilon}):\varepsilon<\text{
otp}(C_{\delta})\times\alpha^{*}\rangle$ where
$\langle\beta_{\delta,\varepsilon}:\varepsilon<\text{ otp}(C_{\delta})\rangle$
list $C_{\delta}*\alpha^{*}$ in increasing order so $F_{\delta}$ is a function
from ${}^{(C_{\delta})}\theta$ into $\theta$. Hence for $\mathaccent
28694{F}=\langle F_{\delta}:\delta\in S\rangle$ there is a diamond sequence
$\langle c_{\delta}:\delta\in S\rangle$. It is easy to compute $\langle
f^{*}_{\delta}:\delta\in S\rangle$ as required (with $\alpha^{*}$). To have
“every $\alpha^{*}<\chi$” we just have to partition $S$ to $\langle
S_{\alpha}:\alpha<\chi\rangle$ such that each $\mathaccent
28694{C}\restriction S_{\alpha}$ has the $\theta$-Md-property which follows
from ?(3) below.

2), 3), 4) Left to the reader. $\square_{?}$

###### Definition 1.10 Definition

1) Let $(\lambda,\mathaccent 28694{C})$ or $\mathaccent 28694{C}$ have the
$(\mathchar 7\relax,\theta)$-MDℓ-property if $\ell=0$ and in ?(1) we demand
$\alpha^{*}_{\delta}=1$ and similar $\ell=2$ with ?(2).

2) Assume $\lambda>\kappa$ are regular and $\mathaccent 28694{C}=\langle
C_{\delta}:\delta\in S\rangle$ is a $(\lambda,\kappa,\chi)$-MD-parameter with
the $(\mathchar 7\relax,\theta)$-MD-property. Let
ID${}_{\lambda,\kappa,\chi,\mathchar 7\relax,\theta}(\mathaccent
28694{C})=\delimiter 69624584A\subseteq\lambda:\mathaccent
28694{C}\restriction(S\cap A)$ does not have the $(\mathchar
7\relax,\theta)$-MD-property$\delimiter 86405897$. If $\mathchar
7\relax=\theta$ we may omit it.

###### \stag{d.12b} Claim

Let $\mathaccent 28694{C},\lambda,\kappa,\chi,\theta$ be as above in
Definition and $J=\text{ ID}_{\lambda,\kappa,\chi,\theta}(\mathaccent
28694{C})$.

1) $J$ is an ideal on $\lambda$ such that $\lambda\delimiter 2548495S\in
J,\emptyset\notin J$.

2) $J$ is $\lambda$-complete and even normal.

3) If $S\subseteq\lambda$ and $S\neq\emptyset$ mod $J$,

then $J$ can be partitioned to $\theta$ subsets and $\neq\emptyset$ mod $J$.

###### Demonstration Proof

Straight as the sequence $\langle F_{\delta}:\delta\in S\rangle$ is not
required to be nicely definable.

$*\qquad*\qquad*$

Well, are there good $(\lambda,\kappa,\kappa^{+})$-parameters? (on
$I[\lambda]$ see [ Sh:420, $\mathchar 632\relax$1]).

###### \stag{d.13} Claim

1) If $S$ is a stationary subset of the regular cardinal $\lambda$ and $S\in
I[\lambda]$ and $(\forall\delta\in S)[{\text{\tenrm cf\/}}(\delta)=\kappa]$,

then

2) If $\kappa={\text{\tenrm cf\/}}(\kappa),\kappa^{+}<\lambda={\text{\tenrm
cf\/}}(\lambda)$, then there is a stationary $S\in I[\lambda]$ with
$(\forall\delta\in S)[{\text{\tenrm cf\/}}(\delta)=\kappa]$.

? Sh:420 ?

3) In part (1), moreover, we can find a good $(S\cap
E,\kappa,\kappa^{+})$-parameter and a $\lambda$-complete filter $D$ on
$\lambda$ such that: for every club $E^{\prime}$ of $\lambda$, the set
$\delimiter 69624584\delta\in S\cap E:C_{\delta}\subseteq E^{\prime}\cap
E\delimiter 86405897$ belongs to $D$.

###### Demonstration Proof

1) By the definition of $I[\lambda]$ see [ Sh:420, $\mathchar 632\relax$1].

? Sh:420 ?

2) By [ Sh:420, $\mathchar 632\relax$1].

? Sh:420 ?

3) By [ Sh:g, III]. $\square_{?}$

? Sh:g ?

###### Definition 1.11 Definition

Let $\mathaccent 28694{C}$ be a $(\lambda,\kappa,\chi)$-parameter

We can strengthen 1.4(2) as follows.

###### Definition 1.12 Definition

We say the $\lambda$-colouring $\mathbb{F}$ is $(S,\kappa,\chi)$-good

if:

We can sum up

###### Demonstration 1.13 Conclusion

If $\lambda=\text{cf}(2^{\mu})$ and we let $\mathchar 2\relax=:\delimiter
69624584\theta=\text{cf}(\theta)$ and
$(\forall\alpha<2^{\mu})(|\alpha|^{<\theta>_{\text{tr}}}<2^{\mu})\delimiter
86405897$

then

then there is a good${}^{+}\mskip 3.0mu(\lambda,\kappa,\kappa^{+})$-MD-
parameter $\langle C_{\delta}:\delta\in S\rangle$

” $(d)$” for $\theta,S,\mathaccent 28694{C}$ as above and $\mathchar
7\relax\leq 2^{\mu}$, if $\mathbb{F}$ is a $(\mathaccent
28694{C},\theta)$-colouring and $D=D_{\lambda}$ (or $D$ is a
$\mu^{+}$-complete filter on $\lambda$ extending $D_{\lambda}$, and lastly
$S\in D^{+}$ then there is a middle diamond for $\mathaccent 28694{C}$ and
there is an $D-\mathbb{F}$-Wd-sequence; see Definition 1.4(3A).

###### Demonstration Proof

Clause (a) holds by [ Sh:460], clause (b), (c) holds by ? and clause (d) by
the major claim ?. $\square_{1.13}$

Note

1.14 Question: 1) Assume $\lambda$ is a weakly inaccessible (i.e. a limit
regular uncountable cardinal but not strong limit) and $\langle
2^{\theta}:\theta<\lambda\rangle$ is not eventually constant.

? Sh:460 ?

Does $\lambda$ have the weak diamond? Other consequences? In this case
cf$(2^{<\lambda})=\lambda$ as $\langle 2^{\theta}:\theta<\lambda\rangle$
witness hence $2^{\lambda}>2^{<\lambda}=:\mathchar 6\relax\delimiter
696245842^{\theta}:\theta<\lambda\delimiter 86405897$. If
$\lambda=2^{<\lambda}$ then $\lambda=\lambda^{<\lambda}$, i.e. as in [
Sh:755].

? Sh:755 ?

2) Consider generalizing [ Sh:f, AP,$\mathchar 632\relax$3] to our present
context, i.e. there we have 2 colours; moreover consider $\langle
c_{\eta}:\eta\in\lambda^{+n}\times\lambda^{+(n-3)}\lambda\mathinner{\ldotp\ldotp\ldotp}\mskip
3.0mu\times\lambda\rangle$ where $c_{\eta}\in\theta$.

? Sh:f ?

###### \stag{d.20} Claim

1) In the proof of ? instead of $f:\lambda\rightarrow\theta$ and
$f_{\delta}:C_{\delta}\rightarrow\theta$ we can use
$f:[\lambda]^{<\aleph_{0}}\rightarrow\theta$ and
$f_{\delta}:[C_{\delta}]^{<\aleph_{0}}\rightarrow\theta$ (so get the
$\theta$-Mdℓ-property for $\ell=0,1,2$).

###### Demonstration Proof

The same.

###### \stag{d.n.21} Claim

Assume

1) Setp$(\mu,\mu,\theta,2^{\mu})$ hence if $2^{\mu}$ is a regular cardinal
then Sep$(\mu,\theta)$, that is clause (f) of ?.

2) Even if $2^{\mu}$ is singular still in Claim 1.4 we can replace clause (f)
in the assumption by $\circledast_{\mu,\theta}$.

###### Demonstration Proof

1) Clearly ${}^{\mu}\theta$ has cardinality $2^{\mu}$, so let
$\langle\nu^{*}_{\alpha}:\alpha<2^{\mu}\rangle$ be a list without repetitions
of ${}^{\mu}\theta$. For $\alpha\leq 2^{\mu}$, let $\mathchar
0\relax_{\alpha}=\delimiter 69624584\nu_{\beta}:\beta<\alpha\delimiter
86405897$ and we choose $\mathaccent 28694{f}^{\alpha}$ by induction on
$\alpha\leq 2^{\mu}$ such that

For clause (d) it is enough to consider the case $\alpha=\beta+1$, and the non
trivial case in the induction is $\alpha=\beta+1$ and we can choose $\langle
f^{\beta+1}_{\varepsilon}(\nu_{\beta}):\varepsilon<\mu\rangle\in{}^{\mu}\theta$
such that it holds by the assumption $(*)$.

2) In the proof of 1.4, now
$\rho\in{}^{\mu}\theta\Rightarrow|\text{Sol}_{\rho}|<2^{\mu}$ do not suffice.
However, before choosing $\langle f_{\varepsilon}:\varepsilon<\mu\rangle$ we
are given $\mathbb{F}$ and we can choose cd,
$\langle\text{cd}_{\varepsilon}:\varepsilon<\mu\rangle$, define
$T,\langle\eta_{\alpha}:\alpha<2^{\mu}\rangle,T_{<\alpha}$ and
$\langle{\Cal{P}}_{\delta}:\delta\in S\rangle$. Note that
$\alpha<\lambda\Rightarrow|h(\alpha)|^{<\kappa>_{\text{tr}}}<2^{\mu}$ hence
$\alpha<\lambda\Rightarrow|\cup\delimiter 69624584{\Cal{P}}_{\delta}:\delta\in
S\cap\alpha\delimiter 86405897|<2^{\mu}$. We can also choose
$\langle\nu_{\eta,\varepsilon}:\eta\in{\Cal{P}}_{\delta},\varepsilon<\mu\rangle$
and $\rho_{\eta}\in{}^{\mu}\theta$ for $\eta\in{\Cal{P}}_{\delta},\delta\in
S$.

Now we can phrase our demands on $\langle
f_{\varepsilon}:\varepsilon<\mu\rangle$, it is

For this it is enough that for the list
$\langle\nu^{*}_{\alpha}:\alpha<2^{\mu}\rangle$ chosen in the proof of part
(1) for each $\delta\in S$, for some $\gamma<2^{\mu}$ we have $\delimiter
69624584\rho_{\eta}:\eta\in\mathbb{P}_{\delta}\delimiter
86405897\subseteq\delimiter 69624584\nu^{*}_{\alpha}:\alpha<\gamma\delimiter
86405897$ which is easy. $\square_{?}$

##  $\mathchar 632\relax$1 Upgrading to larger cardinals 999done March 2001

In $\mathchar 632\relax$1 we get the $\theta$-middle diamond property for
regular cardinals $\lambda$ of the form cf$(2^{\mu})$ on “most” $S\subseteq
S^{\lambda}_{\kappa}$ for “many” $\kappa=\text{ cf}(\kappa)$ (and $\theta$).
We would like to prove Theorem ?. For this we first give an upgrading,
stepping up claim (for MDℓ). Then we prove that it applies to $\lambda_{2}$
(for not few $\lambda_{1}$), when $\lambda_{2}$ is successor of regular, then
when $\lambda_{2}$ is successor of singular and lastly when $\lambda_{2}$ is
weakly inaccessible. We can upgrade each of the variants from $\mathchar
632\relax$1, but we can concentrate on the one from 1.4(5A) because of Claim
?.

The main result is ?.

Question: What occurs to $\lambda_{2}$ strongly inaccessible, does it have to
have some Md-property?

Possibly combining $\mathchar 632\relax$1 and $\mathchar 632\relax$2 we may
eliminate the $\lambda_{2}>2^{\lambda_{1}+\mathchar 7\relax+\theta}$ in ?, but
does not seem urgent now.

###### \stag{x.20} Upgrading Claim

1) Assume

Then for some stationary $S_{2}\subseteq S^{\lambda_{2}}_{\kappa}$, there is
$\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S_{2}\rangle$, a good
$(\lambda_{2},\kappa,\chi)$-Md-parameters which has the
$(D_{\lambda_{2}},\mathchar 7\relax,\theta)$-Md-property.

2) We can in part (1) replace clause (c) by any of the following clauses

3) Assume $\ell\in\delimiter 696245840,2\delimiter 86405897$

Then some $(\lambda,\kappa,\chi)$-MDℓ-parameter has the $\theta$-Mdℓ-property.

###### Demonstration Proof

1) Clearly if clause $(c)$ holds then clause $(c)^{\prime}$ of part (2) holds
by [ Sh:351, $\mathchar 632\relax$4], so it is enough to prove part (2).

? Sh:351 ?

2) Trivially clause $(c)^{\prime}$ implies clause $(c)^{\prime\prime}$ and
clause $(c)^{\prime\prime}$ implies clause $(c)^{\prime\prime\prime}$; so
assume $(c)^{\prime\prime\prime}$; and let $\mathaccent 28694{C}^{2}=\langle
C^{2}_{\delta}:\delta\in S_{2}\rangle$ be as there.

Let $\mathaccent 28694{h}=\langle h_{\delta}:\delta\in S_{2}\rangle$ be such
that $h_{\delta}$ is an increasing continuous function from $\lambda_{1}$ onto
$C^{2}_{\delta}$. Let $S=\delimiter 69624584h_{\delta}(i):\delta\in S_{2}$ and
$i\in S_{1}\delimiter 86405897$ and if $\alpha\in
S,\alpha=h_{\delta}(i),\delta\in S_{2}$ and $i\in S_{1}$ then we let
$C^{*}_{\alpha}=\delimiter 69624584h_{\delta}(j):j\in C^{1}_{\delta}\delimiter
86405897$; and let $\mathaccent 28694{C}^{*}=\langle C^{*}_{\alpha}:\alpha\in
S\rangle$. Why is $C^{*}_{\alpha}$ (hence $\mathaccent 28694{C}^{*}$) well
defined? By clause $(\zeta)$ of the assumption $(c)^{\prime\prime\prime}$. We
should check the conditions for “$\mathaccent 28694{C}$ is a
$(\lambda,\kappa,\theta)$-Md-property.

The main point is: assume that $\mathbb{F}$ is a $(\mathaccent
28694{C}^{*},\mathchar 7\relax,\theta)$-colouring, find a $\mathaccent
28694{c}$ as there.

For each $\delta\in S_{2}$ we define a $(\mathaccent
28694{C}^{1},\kappa,\theta)$-colouring $\mathbb{F}_{\delta}$ by
$\mathbb{F}_{\delta}(\eta)=\mathbb{F}(\eta\circ h_{\delta})$. As 101010only
place this is used, if $\mathbb{F}$ is nicely defined this is not necessary
$\lambda_{2}>2^{\lambda_{1}+\mathchar 7\relax+\theta}$, for some stationary
$S^{*}_{2}\subseteq S_{2}$ we have $\delta\in
S^{*}_{2}\Rightarrow\mathbb{F}_{\delta}=\mathbb{F}_{*}$, so $\mathbb{F}_{*}$
is $(\mathaccent 28694{C}^{1},\mathchar 7\relax,\theta)$-colouring, hence it
has a $(D_{\lambda},\mathchar 7\relax,\theta)$-Md-sequence $\mathaccent
28694{c}^{1}=\langle c^{1}_{i}:i\in S_{1}\rangle$.

We define $\mathaccent 28694{c}^{*}=\langle c^{*}_{\alpha}:\alpha\in S\rangle$
by $\delta\in S_{2}\wedge i\in S_{1}\Rightarrow
c^{*}_{h_{\delta}(i)}=c^{1}_{i}$. By clause $(\zeta)$ of
$(c)^{\prime\prime\prime}$ it is well defined. As our colourings deal with
$\eta\in{}^{(\lambda_{2})}\mathchar 7\relax$ only we clearly are done.

3) Similar only easier (or see the proof of ?). $\square_{?}$

###### \stag{x.21a} Claim

We can strengthen the conclusion of ?(1) and ?(2) to: we can find a sequence
$\langle S_{2,i}:i<\lambda_{2}\rangle$ of pairwise disjoint stationary subsets
of $\lambda_{2}$ and for each $i<\lambda_{2}$ a sequence $\langle
C^{i}_{\delta}:\delta\in S_{2,i}\rangle$ which is a
$(\lambda_{2},\kappa,\chi)$-Md-parameter having the
$(D_{\lambda_{2}},\mathchar 7\relax,\theta)$-Md-property.

###### Demonstration Proof

By the proof of ?, without loss of generality

clause $(c)^{\prime\prime}$ of ?(2) and clauses (a), (b) of ?(1) holds. For
some $\zeta<\lambda_{1}$ for unboundedly many $\gamma<\lambda_{2}$ the set
$S_{2,\gamma}=:\delimiter 69624584\delta\in S_{2}:\gamma$ is the $\zeta$-th
member of $C^{2}_{\delta}\delimiter 86405897$ is stationary. $\square_{?}$

2.1 Conclusion: 1) If $\mu$ is a strong limit, $\lambda$ is a successor
cardinal, $\lambda=\text{ cf}(\lambda)>2^{\mu}$, moreover
$\lambda>2^{2^{\mu}}$ (superfluous for nice colouring, e.g., for
MD${}_{\ell},\ell\in\delimiter 696245840,2\delimiter 86405897$), then for
every large enough regular $\kappa<\mu$, some good${}^{+}\mskip
3.0mu(\lambda,\kappa,\kappa^{+})$-Md-parameter $\mathaccent 28694{C}$ has the
$(2^{\mu},\theta)$-Md-property for every $\theta<\mu$.

2) We can add: $\mathaccent 28694{C}$ has the $\theta$-MDℓ-property for
$\ell=\delimiter 696245840,2\delimiter 86405897$.

3) We can find a sequence $\langle S_{i}:i<\lambda\rangle$ of pairwise
disjoint stationary subsets of $S^{\lambda}_{\kappa}$ such that for each
$i<\lambda$ there is $\mathaccent 28694{C}^{i}=\langle
C^{i}_{\delta}:\delta\in S\rangle$ as in parts (1) and (2).

###### Demonstration Proof

1) Let $\kappa_{0}<\mu$ be large enough such that $\theta\in\delimiter
69624584\lambda,2^{\mu}\delimiter
86405897\and\alpha<\theta\and\kappa_{0}\leq\kappa=\text{
cf}(\kappa)<\mu\Rightarrow|\alpha|^{[\kappa]}<\theta$, see Definition 1.2,
exists by [ Sh:460].

? Sh:460 ?

By ?, for any regular $\kappa\in[\kappa_{0},\mu)$ and $\theta<\mu$ we have the
result for cf$(2^{\mu})$. If $\lambda$ is a successor of regular we can apply
?(1), i.e. with clause $(c)$ there is a square as required by [ Sh:351,
$\mathchar 632\relax$4]. In the case $\lambda$ is a successor of a singular
cardinal, by ? below, clause $(c)^{\prime\prime}$ of ?(2) holds for
$\lambda_{2}=\lambda,\lambda_{1}=\text{cf}(2^{\mu})$ so we can apply ?.

? Sh:351 ?

2) Follows by ?.

3) In the proof before applying ? we use ?. $\square_{2.1}$

Generalizing [ EK]:

###### \stag{x.23} Claim

1) Assume

Then there is $\mathaccent 28694{F}$ such that:

then we can find $\mathaccent 28694{E}$ such that

then $f\subseteq F_{\varepsilon}$ for some $\varepsilon<\mu$. 2) We can
strengthen clause $(\gamma)^{-}$ to $(\gamma)$ below if $(d)_{\ell}$ for some
$\ell\in\delimiter 696245841,\dotsc,5\delimiter 86405897$ below holds

then we can find $\mathaccent 28694{u}=\langle u_{i}:i<\kappa\rangle$ such
that

” $(d)_{1}$” there is a sequence $\mathaccent 28694{f}=\langle
f_{\alpha}:\alpha<\lambda\rangle$ of pairwise distinct functions from $\mu$ to
$\mu$ such that: if $u\in[\lambda]^{\leq\theta}$ then we can find sequences
$\mathaccent 28694{u}=\langle u_{i}:i<\kappa\rangle$ and $\langle
B_{i}:i<\kappa\rangle$ such that $u=\cup\delimiter
69624584u_{i}:i<\kappa\delimiter 86405897,B_{i}\in[\mu]^{<\aleph_{0}}$ for
$i<\kappa$ and for each $i<\kappa$ the sequence $\langle
f_{\alpha}\restriction B_{i}:\alpha\in u_{i}\rangle$ is with no repetitions

” $(d)_{2}$” there is a sequence $\mathaccent 28694{f}=\langle
f_{\alpha}:\alpha<\lambda\rangle$ of functions from $\kappa$ to $\mu$ and
ideal $J$ on $\kappa$ such that $\mathaccent 28694{f}$ is $\theta^{+}$-free
(i.e. for $u\in[\lambda]^{\leq\theta}$, we can find a sequence $\mathaccent
28694{a}=\langle a_{\alpha}:\alpha\in u\rangle$ of subsets of $\kappa$ such
that $a_{\alpha}=\kappa$ mod $J$ and $i\in a_{\alpha}\cap
a_{\beta}\and\alpha\neq\beta\Rightarrow f_{\alpha}(i)\neq f_{\beta}(i)$)

” $(d)_{3}$” like $(d)_{2}$ but $a_{\alpha}\in J^{+}$ that is
$a_{\alpha}\notin J$

” $(d)_{4}$” $(\forall\alpha<\mu)(|\alpha|^{\theta}<\mu)$ and
$\lambda<{\text{\tenrm pp\/}}_{J}(\mu)$ where $J$ is an ideal on $\kappa$

” $(d)_{5}$” $\mu=\mu^{\aleph_{0}}<\lambda<\mu^{\kappa}$. 3) Let $\mathaccent
28694{f}=\langle f_{\alpha}:\alpha<\lambda\rangle$ be a sequence of pairwise
distinct members of ${}^{\mu}\mu$. Then in (1) we can have

then for any $f:u\rightarrow\lambda$, we can find a sequence $\langle
u^{\prime}_{i}:i<\kappa\rangle$ of subsets of $u$ with union $u$ such that
$i<\kappa\Rightarrow f_{i}\restriction u^{\prime}_{i}$ is included in some
$F_{\varepsilon}$. 3A) If $\theta={\text{\tenrm cf\/}}(\theta)$ and
$(\forall\alpha<\theta)(|\alpha|^{\kappa}<\theta)$ then in part (3) clause
$(\gamma)^{\prime}$ implies

4) If we weaken the conclusion of part (2), replacing $(\gamma)$ by

then we can weaken the assumption in a parallel way, e.g.

###### Remark Remark

We can replace $\lambda$ (as a domain) by $[\lambda]^{2}$, etc.

###### Demonstration Proof

Should be clear. (Identifying the function $F$ with $\delimiter
69624584(\alpha,f(\alpha)):\alpha\in\text{ Dom}(f)\delimiter 86405897$.)

$\square_{?}$

###### \stag{x.23.9} Claim

1) Assume

Then there is $\mathaccent 28694{F}$ such that

2) Assume

Then there is a sequence $\langle F_{\varepsilon}:\varepsilon<\mu\rangle$ of
functions from $\lambda$ to $\mu$ such that

###### Demonstration Proof

1) Let $\mathaccent 28694{M}=\langle M_{\alpha}:\alpha<\lambda\rangle$ be an
increasing continuous sequence of elementary submodels of
$({\Cal{H}}(\lambda^{+}),\in,<^{*}_{\chi})$, each of cardinality $<\lambda$
satisfying $\mathaccent 28694{M}\restriction(\alpha+1)\in
M_{\alpha+1},\delimiter 2536205M_{0}\delimiter
2536205=\mu,M_{1+\alpha}\cap\lambda\in\lambda$ and $\delimiter
69624584\lambda,\theta,T,S\delimiter 86405897\subseteq M_{0}$. Let
$E=\delimiter 69624584\delta<\lambda:M_{\delta}\cap\lambda=\delta\delimiter
86405897$, clearly a club of $\lambda$. Let $\mathaccent 28694{a}=\langle
a_{\alpha}:\alpha<\lambda\rangle$ witness $S\in I[\lambda]$ and without loss
of generality

$\mathaccent 28694{a}\in M_{0}$.

Now we choose by induction on $\alpha<\lambda$ a function $f_{\alpha}$ from
$\kappa$ to $\mu$ such that:

Clearly possible.

So for every $i<\kappa$ and $u\in M_{0}\cap[\mu]^{\leq\theta}$ and
$g:u\rightarrow\mu$ from $M_{0}$, let $F_{g,i}\in M_{0}$ be such that
$F_{g,i}:\lambda\rightarrow\mu$ and it satisfies: if $f_{\alpha}(i)\in\text{
Dom}(g)$ then $F_{g,i}(\alpha)=g(f_{\alpha}(i))$.

Let $\langle(g_{\varepsilon},i_{\varepsilon}):\varepsilon<\mu\rangle$ list the
set of those pairs $(g,i)$ and let
$F_{\varepsilon}=F_{g_{\varepsilon},i_{\varepsilon}}$. We shall show that
$\langle F_{\varepsilon}:\varepsilon<\mu\rangle$ is as required; so clauses
$(\alpha)$ \+ $(\beta)$ holds trivially.

Now if $\delta\in\text{ acc}(E)\cap S$, let
$\langle\alpha_{\varepsilon}:\varepsilon<\theta\rangle$ list the closure of
$a_{\delta}$ in increasing order so for some club $C$ of $\theta$ we have
$\varepsilon\in C\Rightarrow\alpha_{\varepsilon}\in E$. Now by assumption (e)
clearly $\varepsilon\in C\wedge\text{ cf}(\varepsilon)\in\mathchar
2\relax\Rightarrow\langle\alpha_{\zeta}:\zeta\leq\varepsilon\rangle\in
M_{\alpha_{\varepsilon}+1}$.

Let

$\displaystyle u_{i}=\delimiter 69624584\varepsilon\in C:$
$\displaystyle\text{cf}(\varepsilon)\in\mathchar 2\relax\text{ and}$
$\displaystyle f_{\alpha}(i)\notin\cup\delimiter
69624584\text{Rang}(f(\beta)):\beta\in\delimiter
69624584\alpha_{\zeta}:\zeta<\varepsilon\delimiter 86405897\delimiter
86405897.$

Let $h$ be a function from $\delimiter
69624584\alpha_{\varepsilon}:\varepsilon<\theta\delimiter 86405897$ to $\mu$.
So we can find a sequence $\langle w_{j}:j<\kappa\rangle$ of subsets of $\mu$
of cardinality $\leq\theta$ such that Rang$(h)\subseteq\displaystyle\mathchar
4947\relax\displaylimits_{j<\kappa}w_{j}$ and $\delimiter
69624584w_{j}:j<\kappa\delimiter 86405897\subseteq M_{0}$. Hence for
$i,j<\kappa$ there is $g:u_{i}\cap w_{j}\rightarrow\mu$ which belong to
$M_{0}$ and satisfies $\varepsilon\in u_{i}\cap w_{j}\Rightarrow
g(f_{\alpha_{\varepsilon}}(i))=h(\alpha_{\varepsilon})$, see the choice of
$u_{i}$. So easily $F_{g,i}$ extends $h\restriction(u_{i}\cap w_{j})$. So
$\langle u_{i}\cap w_{j}:i,j<\kappa\rangle$ and
$\langle\alpha_{\varepsilon}:\varepsilon\in E\rangle$ are as required (modulo
some renaming).

2) Easy by now and if $\mu=\mu^{\kappa}$ by [ EK], and if not (so
cf$(\mu)=\kappa$) then known.

? EK ?

$\square_{?}$

###### \stag{x.2.9} Claim

1) Assume

Then we can find $(\mathaccent 28694{C}_{i},A_{i},S^{*}_{i})$ for $i<\mu$ such
that

2) Assume $\alpha<\mu\Rightarrow|\alpha|^{\theta}<\mu,T_{\theta}=\delimiter
69624584\delta<\theta:\text{cf}(\delta)\neq\kappa\delimiter 86405897$ and we
let $\mathbb{P}=\delimiter 69624584\mathaccent 28694{A}:\mathaccent
28694{A}=\langle A_{i}:i<\kappa\delimiter 86405897$ is an increasing sequence
of sets with union $T_{\theta}\delimiter 86405897$. Then for some ${\Cal{P}}$
all the assumptions hence the conclusion of part (1) holds.

3) Assume $\mu_{*}$ is strong limit $<\mu,\kappa=\mskip 3.0mu{\text{\tenrm
cf\/}}(\mu)<\mu,\lambda=\mu^{+}$, then for some $\theta_{0}<\mu_{*}$, for any
regular $\theta\in(\theta_{0},\mu_{*})$ the set $T_{\theta}$ include
$\delimiter 69624584\delta<\theta:\text{cf}(\delta)\geq\theta_{1}$ or $\delta$
nonlimit$\delimiter 86405897$ and for $\mathbb{P}=\delimiter
69624584\mathaccent 28694{A}:\mathaccent 28694{A}$ is a sequence of
$<\theta_{0}$ subsets of $\theta$ with union $T_{\theta}$ such that
$\delimiter 69624584A_{\zeta}:\zeta<\ell g(\mathaccent 28694{A})\delimiter
86405897$ is closed under finite unions$\delimiter 86405897$.

Then for some ${\Cal{P}}$ the assumptions hence the conclusion of part (1)
holds.

4) In (1) (and (2), (3), if $\mathaccent 28694{A}=\langle
A_{\varepsilon}:\varepsilon<\varepsilon(*)\rangle\in\mathbb{P}$ we can replace
it by $\mathaccent 28694{A}^{\prime}=\langle
A^{\prime}_{\varepsilon}:\varepsilon<\varepsilon(*)\rangle,A^{\prime}_{\varepsilon}=A_{\varepsilon}\cup\delimiter
69624584\delta<\theta:\text{cf}(\delta)>\aleph_{0}$ and
$A_{\varepsilon}\cap\delta$ is stationary in $\delta\delimiter 86405897$.

###### Remark 2.2 Remark

If $\lambda$ is (weakly) inaccessible we have weaker results there is
$\langle{\Cal{P}}_{\alpha}:\alpha<\lambda\rangle,{\Cal{P}}_{\alpha}\subseteq[\alpha]^{<\theta},|{\Cal{P}}_{\alpha}|<\lambda$
such that for every stationarily many $\delta\in S^{\lambda}_{\theta}$ for
some increasing continuous sequence
$\langle\alpha_{\varepsilon}:\varepsilon<\alpha\rangle$ with limit
$\delta,\varepsilon<\theta\and\text{ cf}(\varepsilon)\in\mathchar
2\relax_{\lambda}\Rightarrow\delimiter
69624584\alpha_{\zeta}:\zeta<\varepsilon\delimiter
86405897\in{\Cal{P}}_{\alpha_{\varepsilon}}$ (see [ Sh:420, $\mathchar
632\relax$1]).

? Sh:420 ?

###### Demonstration Proof

There is $B$ as in clause (e), that is $B\in I[\lambda]$ and $B\subseteq
S^{\lambda}_{\theta}$ is a stationary subset of $\lambda$, see Claim ?. So let
$B$ be such a set; clearly omitting from $B$ a nonstationary subset of
$\lambda$ is O.K. (see clause (e)).

We can find $\mathaccent 28694{a}=\langle a_{\alpha}:\alpha<\lambda\rangle$
witnessing $B\in I[\lambda]$ so (possibly omitting from $B$ a nonstationary
set) we have: $\alpha\in
B\Rightarrow\alpha=\mathop{sup}\displaylimits(a_{\alpha})\and\theta=\text{
otp}(a_{\alpha})$ and $[\beta\in a_{\alpha}\Rightarrow a_{\beta}=\beta\cap
a_{\alpha}]$ and $[\alpha\in\lambda\delimiter 2548495B\Rightarrow\text{
otp}(a_{\alpha})<\theta]$.

Let $\chi=\lambda^{+}$ and $\mathaccent 28694{M}=\langle
M_{\alpha}:\alpha<\lambda\rangle$ be an increasing continuous sequence of
elementary submodels of $({\Cal{H}}(\chi),a,<^{*}_{\chi})$ to which
$\delimiter 69624584\mathaccent 28694{a},\mu,\mathbb{P},{\Cal{P}}\delimiter
86405897$ belongs satisfying $\mathaccent 28694{M}\restriction(\alpha+1)\in
M_{\alpha+1}$ and $\delimiter 2536205M_{\alpha}\delimiter 2536205<\lambda$.

For any $\alpha<\lambda$ let $\langle b_{\alpha,\zeta}:\zeta<\mu\rangle$ list
$M_{\alpha+1}\cap[\alpha]^{<\theta}$.

Let $\langle\lambda_{i}:i<\kappa\rangle$ be an increasing sequence of regular
cardinals with limit $\mu,\lambda_{\varepsilon}>\theta$ with $\lambda=\text{
tcf}(\displaystyle\mathchar
4945\relax\displaylimits_{i<\kappa}\lambda_{i},<_{J^{\text{bd}}_{\kappa}})$
and let $\mathaccent 28694{f}=\langle f_{\alpha}:\alpha<\lambda\rangle$ be an
increasing sequence in $(\displaystyle\mathchar
4945\relax\displaylimits_{i<\kappa}\lambda_{i},<_{J^{\text{bd}}_{\kappa}})$
cofinal in it and obeying $\mathaccent 28694{a}$ (i.e. $i<\kappa\and\beta\in
a_{\alpha}\Rightarrow f_{\beta}(i)<f_{\alpha}(i)$ and $\zeta<\lambda_{i}\wedge
i<j<\kappa\wedge\beta\in b_{\alpha,\zeta}\Rightarrow
f_{\beta}(j)<f_{\alpha}(j)$ (see [ Sh:g]).

? Sh:g ?

Now for any $x=(h_{1},h_{2},A,i)=(h^{x}_{1},h^{x}_{2},A^{x},i^{x})$ where
$h_{1}$ is a partial function from $\mu$ to $\theta,h_{2}$ a function from $A$
to $\mu$ where $T^{-}_{\theta}\subseteq A\subseteq T^{+}_{\theta}$ and
$i<\kappa$, we define

$\displaystyle S_{x}=\biggl{\delimiter 69624584}\alpha<\lambda:$
$\displaystyle\text{we have }f_{\alpha}(i)\in\text{ Dom}(h_{1})\text{ and
}\zeta=:h_{1}(f_{\alpha}(i))\in A\text{ satisfies}$ $\displaystyle(i)\quad
b_{\alpha,h_{2}(\zeta)}\text{ is a subset of }\alpha\text{ of order type
}\zeta,$ $\displaystyle(ii)\quad b_{\alpha,h_{2}(\zeta)}\text{ is unbounded in
}\zeta\text{ if }\zeta\text{ is a limit ordinal}$
$\displaystyle(iii)\quad\text{if }\varepsilon\in A\cap\zeta\text{ and
}\beta\in b_{\alpha,h_{2}(\zeta)}\text{ and }\varepsilon=\text{
otp}(b_{\alpha,\zeta}\cap\beta)$ $\displaystyle\qquad\qquad\mskip 3.0mu\mskip
3.0mu\text{then }\varepsilon=h_{1}(f_{\beta}(i))\and\beta\cap
b_{\alpha,\zeta}=b_{\beta,h_{2}(\varepsilon)}\biggr{\delimiter 86405897}$

and for $\alpha\in S_{x}$ let
$C^{x}_{\alpha}=b_{\alpha,h_{2}(h_{1}(f_{\alpha}(i))}$.

Let $\langle X_{\varepsilon}:\varepsilon<\varepsilon^{*}\leq\mu\rangle$ list
the set of $x\in M_{0}$ which are as above and $S^{*}_{x}=:\delimiter
69624584\delta\in S_{x}:\text{cf}(\delta)=\theta\delimiter 86405897$ is
stationary, and let $\mathaccent 28694{C}_{\varepsilon}=:\langle
C^{x_{\varepsilon}}_{\alpha}:\alpha\in
S_{x_{\varepsilon}}\rangle,A_{\varepsilon}=:A^{x_{\varepsilon}}$ and
$S_{\varepsilon}=:S_{x_{\varepsilon}}$.

We have to check the five clauses of ?(1):

Clause $(\alpha)$: See the demand on $x$ above.

Clause $(\beta)$: Holds trivially by the definition “$\alpha\in S_{x}"$ and of
$C^{x}_{\alpha}$ for $\alpha\in S_{x}$.

Clause $(\gamma)$: Holds by the choice of $\langle
x_{\varepsilon}:\varepsilon<\varepsilon^{*}\rangle$.

Clause $(\delta)+(\varepsilon)$: Let $\delta\in B$, so let
$\langle\alpha_{\zeta}:\zeta\leq\theta\rangle$ list the closure of
$a_{\delta}$ in increasing order.

Now for every $\zeta\in T^{+}_{\theta}$, for some $\xi(\zeta)<\mu$ we have
$\delimiter 69624584\alpha_{\xi}:\xi<\zeta\delimiter
86405897=b_{\alpha,\xi(\zeta)}$ and let $j(\zeta)=\text{ Min}\delimiter
69624584j<\kappa:\lambda_{j}>\xi(\zeta)\delimiter 86405897$.

For each $j$ let $A^{*}_{j}=\delimiter 69624584\zeta:\zeta\in T^{-}_{\theta}$
or $\zeta\in T_{\theta}\and j(\zeta)<\zeta\delimiter 86405897$ and apply
clause $(e)$ of the assumption to
$\langle\alpha^{\prime}_{\varepsilon}=:f_{\alpha_{\varepsilon}}(j):\varepsilon\in
A^{*}_{j}\rangle$ pedantically complete in
$\langle\alpha^{\prime}_{\varepsilon}:\varepsilon<\theta\rangle\in{}^{\theta}\mu$,
getting $\mathaccent 28694{A}$. The rest should be clear.

2), 3), 4) Left to the reader. $\square_{?}$

###### \stag{x.24} Claim

1) If $\lambda$ is regular and not strongly inaccessible and $\mu<\lambda$ is
strong limit singular of uncountable cofinality,

then for every large enough regular $\kappa<\mu$, for every $\mathchar
7\relax,\theta<\mu$, some good $(\lambda,\kappa,\kappa^{+})$-parameters
$\mathaccent 28694{C}$ has the $(\mathchar 7\relax,\theta)$-Md-property for
$\ell=0,1,2$.

2) Moreover, we can find such $\mathaccent 28694{C}^{\alpha}$ for
$\alpha<\lambda$ satisfying: $\langle{\text{\tenrm Dom\/}}(\mathaccent
28694{C}^{\alpha}):\alpha<\lambda\rangle$ are pairwise disjoint.

###### Demonstration Proof

1) Let $\lambda_{1}<\lambda$ be such that $\lambda\leq 2^{\lambda_{1}}$ and
let $\kappa_{0}<\mu$ be large enough such that

then $\alpha<\lambda\Rightarrow\text{
cov}(|\alpha|,\mu,\mu,\kappa_{0})<\lambda$

” ${{}}(ii)$” if $\lambda$ is inaccessible, then
cov$(\lambda_{1},\mu,\mu,\kappa_{0})=\lambda_{1}$.

(Exist by [ Sh:460].)

? Sh:460 ?

Note that $\lambda_{1}$ is needed only if $\lambda$ is (weakly) inaccessible.

Let $\mu_{*}\in(\kappa_{0},\mu)$ be strong limit; this is possible as $\mu$ is
strong limit of uncountable cofinality. Let
$\kappa_{1}=\kappa_{1}[\mu_{*}]<\mu_{*}$ be $>\kappa_{0}$ and such that
$\alpha<2^{\mu_{*}}\Rightarrow\text{
cov}(|\alpha|,\mu,\mu,\kappa_{0})<2^{\mu_{*}}$.

Let $\kappa\in\text{ Reg}\cap\mu_{*}\delimiter 2548495\kappa_{1}$ and we shall
prove the result for $\kappa$ and every $\mathchar 7\relax,\theta<\mu_{*}$, so
without loss of generality$\mathchar 7\relax=\theta$. By Fodor’s Lemma on
$\mu$ this is enough.

Let $\lambda_{*}=\text{ cf}(2^{\mu_{*}})$. By ?(1) for every stationary
$S\subseteq\delimiter
69624584\delta<\lambda_{*}:\text{cf}(\delta)=\kappa\delimiter 86405897$ from
$I[\lambda_{*}]$ (and there are such, see ?(2)) there are $\mathaccent
28694{C}^{S}=\langle C^{S}_{\delta}:\delta\in S\rangle,D$ such that
$\mathaccent 28694{C}^{S}$ is a weakly good${}^{+}\mskip
3.0mu(\lambda_{*},\kappa,\kappa^{+})$-Md-parameter, $D$ a
$\lambda_{*}$-complete filter on $\lambda_{*}$ containing the clubs of
$\lambda_{*}$ and by ? + ?, the pair $(\mathaccent 28694{C}^{S},D)$ has the
$\theta$-Md-property.

For the case $\lambda$ is (weakly) inaccessible we add “each $C_{\delta}$ is a
club of $\delta$”. This is O.K. as we can use clause $(\beta)$ of ?(1). Fix
$S_{1}\subseteq S^{\lambda_{*}}_{\kappa}$ and $\mathaccent 28694{C}=\langle
C_{\delta}:\delta\in S\rangle=\mathaccent 28694{C}^{S}$.

Note that we actually have

Now

Case 1: $\lambda$ is a successor of regulars.

We apply ?(1) with $(\lambda,\lambda_{*})$ here standing for
$(\lambda_{2},\lambda_{1})$ there; there is suitable partial square (in ZFC by
[ Sh:351, $\mathchar 632\relax$4]). Note that we do not “lose” any
$\kappa<\mu$ because of “the lifting to” $\lambda$.

Case 2: $\lambda$ is a successor of a singular.

? Sh:351 ?

The proof is similar only now we use ?(3) to get the appropriate partial
square using $\boxtimes$ to derive the $\mathaccent 28694{C}^{\prime}$ used as
input for it (recalling each sequence $\mathaccent 28694{A}\in\mathbb{P}$ is
closed under union of two!)

Case 3: $\lambda$ is (weakly) inaccessible.

Really the proof in this case covers the other cases (but $\lambda_{1}$ in
this case contributes to increasing $\kappa_{0}$). This time we use ? + ?.

Let $S_{2}$ be a stationary subset of $S^{\lambda}_{\lambda_{*}}$ from
$I[\lambda]$ it exists, see ?(2). Let $\langle
F_{\varepsilon}:\varepsilon<\lambda_{1}\rangle$ be as in ? only we allow in
the range of $F_{\varepsilon}$ also $\delimiter
69624584(\zeta,e):\zeta<\lambda_{*}$ and $e\subseteq\kappa\delimiter 86405897$
with $(\lambda,\lambda_{1},\lambda_{*},\kappa)$ here standing for
$(\lambda,\mu,\theta,\kappa)$ there. Possibly replacing $S_{2}$ by its
intersection with some club of $\lambda$, there is a good $\lambda_{*}$-Md-
parameter $\mathaccent 28694{C}^{2}=\langle C^{2}_{\delta}:\delta\in
S_{2}\rangle$. Clearly

For this $\varepsilon$ let

We continue as in ?, ?.

2) Similarly using Claim ? when using Claim 2.1. $\square_{?}$

We now state another version of the upgrading of middle diamonds.

###### \stag{x.20.9} Claim

Assume

Then $\mathaccent 28694{C}^{2}$ has the $(\mathchar 7\relax,\theta)$-MDℓ-
property.

###### Remark Remark

1) Is clause (d) of the assumption reasonable? Later in ? we prove that it
holds under reasonable conditions.

2) This + ? are formalized in Definition ?.

###### Demonstration Proof

Similar to ?.

First, assume $\ell=2$ and let $\tau$ be a relational vocabulary of
cardinality $<\mathchar 7\relax$ and by clause (b) of the assumption there is
a sequence $\mathaccent 28694{M}^{1}=\langle M^{1}_{\delta}:\delta\in
S_{1}\rangle,M^{1}_{\delta}$ a $\tau$-model with universe $C^{1}_{\delta}$
such that

then the set of $\delta\in S_{1}$ for which $M^{1}_{\delta}=M\restriction
C^{1}_{\delta}$ is $\neq\emptyset$ mod $D_{\lambda_{1}}$.

Now for every $\delta\in S_{2}$ if $F(\delta)\in S_{1}$ and $\delimiter
69624584F(\alpha):\alpha\in C^{2}_{\delta}\delimiter 86405897\subseteq
C^{1}_{f(\delta)}$ is an unbounded subset of $f(\delta)$, then let
$M^{2}_{\delta}$ be the unique $\tau$-model with universe $C^{2}_{\delta}$
such that $F\restriction C^{2}_{\delta}$ is an isomorphism from
$M^{2}_{\delta}$ onto $M^{1}_{F(\delta)}\restriction\delimiter
69624584F(\alpha):\alpha\in C^{2}_{\delta}\delimiter 86405897$. If $\delta\in
S_{1}$ does not satisfy the requirement above let $M^{2}_{\delta}$ be any
$\tau$-model with universe $C^{2}_{\delta}$.

Now it is easy to check that $\mathaccent 28694{M}^{2}=\langle
M^{2}_{\delta}:\delta\in S_{2}\rangle$ is as required.

Second, assume $\ell=1$ and $\mathbb{F}^{2}$ be a $(\mathaccent
28694{C}^{1},\mathchar 7\relax,\theta)$ colouring. For each $(\mathaccent
28694{C}^{1},\mathchar 7\relax,\theta)$-colouring $\mathbb{F}^{1}$ we ask:
Question on $\mathbb{F}^{1}$: Do we have, for every club $E$ an increasing
continuous function $h:\lambda_{1}\rightarrow\lambda_{2}$ such that

If for some $\mathbb{F}^{1}$ the answer is yes, we proceed as in the proof for
$\ell=1$. Otherwise, for each such $\mathbb{F}^{1}$ there is a club
$E[\mathbb{F}^{1}]$ exemplifying the negative answer. Let $E=\cap\delimiter
69624584E[\mathbb{F}^{1}]:\mathbb{F}^{1}$ is a $(\mathaccent
28694{C}^{1},\mathchar 7\relax,\theta)$-colouring$\delimiter 86405897$. As
there are $\leq 2^{2^{\lambda_{1}}}$ such $\mathbb{F}^{1},E$ is a club of
$\lambda_{2}$ and using (d) of the assumption, there is
$h:\lambda_{1}\rightarrow\lambda_{2}$ as there. Now we can define
$\mathbb{F}^{1}$ by $h,\mathbb{F}^{2}$ naturally and get a contradiction.

The case $\ell=0$ is left to the reader. $\square_{?}$

Trying to apply ?, clause (d) of the assumption look the more serious. By the
following if each $C^{1}_{\delta}$ are stationary subsets of $\delta$ (so
$\kappa>\aleph_{0}$), it helps.

###### \stag{x.20.10} Claim

Is ?, if $2^{\lambda_{1}}<\lambda_{2}$, and we can replace $(d)$ by
$(d)^{-}+(f)$ below,

then we can conclude that for some $\mathaccent 28694{C}^{3}=\langle
C^{3}_{\delta}:\delta\in S\rangle$ has the $\theta$-MDℓ-property where
$C^{3}_{\delta}\in D_{\delta}$

###### Demonstration Proof

Let ${\Cal{C}}=\delimiter 69624584\mathaccent 28694{e}:\mathaccent
28694{e}=\langle e_{\delta}:\delta\in S_{1}\rangle,e_{\delta}$ is a subset of
$C^{2}_{\delta}\delimiter 86405897$.

So $|{\Cal{C}}|\leq 2^{\lambda_{1}}<\lambda$ and for each $\mathaccent
28694{e}\in E$ we define $\mathaccent 28694{C}^{\mathaccent 28694{e}}=\langle
C^{\mathaccent 28694{e}}_{\delta}:\delta\in S^{\mathaccent
28694{e}}_{2}\rangle$ by $S^{\mathaccent 28694{e}}_{\delta}=\delimiter
69624584\delta\in S_{2}:C^{\mathaccent 28694{e}}_{\delta}\in
D_{\delta}\delimiter 86405897$ where $C^{\mathaccent
28694{e}}_{\delta}=\delimiter 69624584\alpha\in
C^{2}_{\delta}:\text{otp}(C^{2}_{\delta}\cap\alpha)\in e_{F(\delta)}\delimiter
86405897$. If for some $\mathaccent 28694{e}\in{\Cal{C}},\mathaccent
28694{C}^{\mathaccent 28694{e}}$ is as required. $\square_{?}$

##  $\mathchar 632\relax$3 Catching more cofinalities 1111112002/4/23

It is clear that in the Major Claim ? a reasonable hope is to weaken the
assumption of “$\alpha<\lambda\Rightarrow|\alpha|^{<\kappa>_{tr}}<\lambda$”.
We look here at sufficient conditions for what is really needed in ?, i.e.,
the $2^{\mu}$-few branches property. The pcf assumptions here are not stated
in an optimal but just shortest way.

###### Definition 4.1 Definition

Let $\lambda>\theta>\aleph_{0}$ be regular.

1) We say that $\mathaccent 28694{C}$ is a $(\lambda,<\theta)$-pseudo square
if

2) We say that $\mathaccent 28694{C}$ is a $(\lambda,<\theta,\kappa)$-pseudo
square if in addition

3) We add “closed” if each $C_{\delta}$ is required to be closed.

4) We replace the pseudo by partial if for $\delta_{1},\delta_{2}\in S$ we
have $\delta_{1}\in C_{\delta_{2}}\Rightarrow
C_{\delta_{1}}=C_{\delta_{2}}\cap\delta_{1}$.

5) We replace pseudo by pseudo0 if instead $S^{2}_{\theta}[\mathaccent
28694{C}]$ we use $S^{0}_{\theta}[\mathaccent 28694{C}]$ where

$\displaystyle S^{0}_{\theta}[\mathaccent 28694{C}]=:\delimiter
69624584\delta<\lambda:$ $\displaystyle\text{cf}(\delta)=\theta\text{ and for
some increasing}$ $\displaystyle\text{continuous sequence
}\langle\alpha_{\delta,\varepsilon}:\varepsilon<\theta\rangle$
$\displaystyle\text{with limit }\delta,\text{the set }\delimiter
69624584\zeta<\theta:\alpha_{\delta,\zeta}\in S\text{ and}$
$\displaystyle\zeta=\mathop{sup}\displaylimits\delimiter
69624584\alpha_{\delta,\varepsilon}:\varepsilon<\delta\text{ and
}\alpha_{\delta,\varepsilon}\in C_{\delta}\delimiter 86405897$
$\displaystyle\text{is a stationary subset of }\theta\delimiter 86405897$

Let pseudo1 mean pseudo.

###### \stag{p.2} Claim

Assume

Then $\mathaccent 28694{C}$ has the weakly $\lambda$-few-branches-property,
see below.

2) If (a) - (d) and (e)μ below, then $\mathaccent 28694{C}$ has the weakly
$\mu$-few-branches property where

We shall prove it below but first:

###### Definition 4.2 Definition

Assume $\mathaccent 28694{C}=\langle C_{\delta}:\delta\in
S\rangle,S\subseteq\lambda$ stationary,
$C_{\delta}\subseteq\delta=\mathop{sup}\displaylimits(C_{\delta})$.

1) Let ${\Cal{C}}[\mathaccent 28694{C}]=\delimiter
69624584C_{\alpha}\cap\beta:\alpha\in S,\beta\in C_{\alpha}\delimiter
86405897$.

2) $T[\mathaccent 28694{C}]=\delimiter 69624584\eta:\eta\in{}^{\Cal{C}}X$ for
some $C\in{\Cal{C}}[\mathaccent 28694{C}]$.

3) We say that $\mathaccent 28694{C}$ has $\lambda$-few-branches properties if
some $\mathaccent 28694{\Cal{F}}$ witness it which means that

4) We say that $\mathaccent 28694{C}$ has the $\mu$-few branches property if
some pair $(h,\mathaccent 28694{\Cal{F}})$ witness it which means

5) We say that $\mathaccent 28694{C}$ has the $h$-few-branches property if $h$
is an increasing continuous function from $\lambda$ into some
$\mu=\mathop{sup}\displaylimits(\text{Rang}(h))$ and some $\mathaccent
28694{\Cal{F}}$ witness it which means that $(\beta),(\gamma)$ above holds.

6) ${\Cal{C}}_{wk}[\mathaccent 28694{C}]=\delimiter 69624584A:A$ is a bounded
subset of some $C_{\delta}\delimiter 86405897$.

7) $T_{wk}[\mathaccent 28694{C}]=\delimiter 69624584\eta:\eta\in{}^{C}\lambda$
for some $C\in{\Cal{C}}_{wk}[\mathaccent 28694{C}]\delimiter 86405897$.

8) In (3), (4) we add weak which means that in $(\gamma)$ we replace
$\eta\restriction C_{\delta}$ by $\eta\restriction C^{\prime}$ for some
unbounded $C^{\prime}\subseteq C_{\delta}$.

###### Demonstration 4.3 Observation

1) If $\mathaccent 28694{C}$ has the $(\lambda,\kappa,\chi)$-Md-parameter and
it has the weak $\mu$-few branches property and $2^{<\chi}<\mu$

then there is $\mathaccent 28694{C}^{\prime}$ such that:

2) If $\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$ is a
$(\lambda,\kappa,\chi)$-Md-parameter with the [weak] $\mu$-few branches
property, then for some $\gamma<\chi,\mathaccent 28694{C}\restriction
S^{[\gamma]}$ has the [weak] $\mu$-few-branches property where
$S^{[\gamma]}=\delimiter 69624584\delta\in
S:\text{otp}(C_{\delta})=\delta\delimiter 86405897$.

###### Demonstration Proof

First we fix $\langle\text{otp}(C_{\delta}):\delta\in S\rangle$ using a
pairing function, trying all possibilities. Second we fix the subset.
$\square_{4.3}$

###### Demonstration Proof of 4.2

Straight by pcf theory. $\square_{4.2}$

On WNR see [ DjSh:562], [ CDSh:571], [ DjSh:691]. This is the case
$(\lambda,\kappa,\kappa)$ below?

###### Definition 4.4 Definition

1) We say $(\lambda,\theta,\kappa)$ has NPr (and write
NPr$(\lambda,\theta,\kappa)$ if:

2) Let Pr$(\lambda,\theta,\kappa)$ be the negation of
WNRef$(\lambda,\theta,\kappa)$.

? DjSh:562 ?

? CDSh:571 ?

? DjSh:691 ?

3) For $D$ a filter on $\kappa$ extending the club filter on $\kappa$ or just
a subset of ${\Cal{P}}(\kappa)$ satisfying: if $A,B\subseteq\kappa,A\mathchar
1\relax B$ is not stationary then $(A_{1}\in D)\equiv(A_{2}\in D)$. We say
that $(\lambda,\theta,\kappa,D)$ has weak nonreflection (also written
WNRef$(\lambda,\theta,\kappa,D)$ if:

Let Pr$(\lambda,\theta,\kappa,D)$ be the negation. We always assume
$A,B\subseteq\kappa\wedge E\subseteq\kappa$ a club $A\cap E=B\cap
E\Rightarrow(A\in D\equiv B\in D)$.

4) Assume that $\mathbb{D}$ is an index-family of filters, ${\Cal{E}}$ a
filter on the index set Dom$(\mathbb{D})$ or on $\mathbb{D}$.

5) Let Dom$(\mathbb{D})=\cup\delimiter
69624584\text{Dom}(D):D\in\mathbb{D}\delimiter 86405897$.

6) We say $\mathbb{D}$ is $({\Cal{E}},\theta)$-dense

if for every function $f$ with domain supp$({\Cal{D}})$ and range of
cardinality $<\theta$ we have

$\delimiter 69624584D\in\mathbb{D}:f\text{ is contant on some }A\in
D\delimiter 86405897\neq\emptyset\text{ mod }E$

7) We say $\mathbb{D}$ is $({\Cal{E}},\theta)$-positive if for every function
$f$ from supp$(\mathbb{D})$ to some bounded subset of $\theta,\delimiter
69624584D:f\text{ is constant on some }A\in D^{+}\delimiter
86405897\neq\emptyset$ mod $D$.

###### Demonstration 4.5 Observation

1) Assume NPr$(\lambda,\theta,\kappa,D)$ and $\lambda$ is regular
$h:\lambda\rightarrow\theta$ and $C_{\delta}$ is a club of $\delta$ of order
type $\kappa$ for $\delta\in S^{\lambda}_{\kappa}$

then the set $R_{h}=\delimiter
69624584\delta<\lambda:\text{cf}(\delta)=\kappa$ and $\delimiter
69624584\text{otp}(\alpha\cap C_{\delta}):\alpha\in C_{\delta}$ and
$h(\alpha)=i\delimiter 86405897$ belong to $D$ for some $i<\theta\delimiter
86405897$ is stationary.

###### Demonstration Proof

If $h:\lambda\rightarrow\theta$ and $E$ is a club of $\lambda$ disjoint to the
set $R_{h}$ then let $h^{\prime}:\lambda\rightarrow\theta$ be defined by
$h^{\prime}(\alpha)=g(g(\alpha))$ where $g:\lambda\rightarrow E$ is increasing
continuous. So if $R_{h}$ is not stationary $E$ is a club disjoint to it we
get that $h^{\prime}$ witness WNRef$(\lambda,\theta,\kappa)$.

###### \stag{p.7} Claim

1) Assume

Then for some $\mathaccent 28694{C}$

2) Assume (a)-(d) and

Then for some $\mathaccent 28694{C}$, $(\alpha),(\beta),(\gamma)$ and
$(\delta)$ $\mathaccent 28694{C}$ has $\mu$-few branches property.

###### Demonstration Proof

Left to the reader.

###### \stag{p.8} Claim

1) In ? we can replace assumption (x) to

2) Moreover, we have the squared middle diamond [here? see $\mathchar
632\relax$6].

###### Demonstration Proof

The same.

For proving ?, for omitting in ? the assumption $S\in I[\lambda]$ which we use
to find a weakly good $(\lambda,\kappa,\chi)$-Md-parameter but strengthen
$|\delta|^{<\text{cf}(\delta)>_{tr}}<\lambda$ to
$|\delta|^{<\text{cf}(\delta)>}<\lambda$ hence for proving the results in the
end of the next section where $\lambda\in I[\lambda]$ is used and using the
squared middle diamond the following helps.

###### \stag{p.8a} Existence Claim

Then we can find $j<\mu$ such that: if $\mathaccent 28694{F}$ is a
$(\mathaccent 28694{C},2^{\mu},\theta)$-colouring then for some $\langle
C_{\delta}:\delta\in S\rangle\in{}^{S}\theta$ we have:

for any $\eta\in{}^{\lambda}(2^{\mu})$ and club $E$ of $\lambda$ for some
increasing continuous sequence $\langle
S_{\gamma}:\gamma\leq\gamma^{*}\rangle$ of members of $E$ we have:

We can now conclude

4.6 Conclusion: 1) If $\lambda=2^{\mu}$ and $\mu_{*}<\mu$ is strong limit,
then for any $\theta<\mu_{*}$ and uncountable regular $\kappa<\mu_{*}$, some
$(\lambda,\kappa,\kappa^{+})$-Md-parameter $\mathaccent 28694{C}$ has the
$\theta$-Md-property provided that (a) or (b) holds

Discussion: 1) We can weaken the pcf demand.

2) If we have the $(\beth_{\omega},\aleph_{\omega})$-property for each
$\lambda$ for some $n(*)<m(*)$ are large enough,
$\langle{\mathfrak{a}}:i<\beth^{+}_{m(*)}\rangle$ is a sequence of finite
${\mathfrak{a}}_{i}\subseteq[\beth_{\omega},\lambda)\cap\text{ Reg}$ then for
some stationary $S\subseteq S^{\beth^{+}_{m(*)}}_{\aleph_{n(*)+1}}$, for each
$\delta\in S$ for some stationary $S^{\prime}\subseteq\delta$, max pcf
$\cup\delimiter 69624584{\mathfrak{a}}_{i}:i\in S^{\prime}\delimiter
86405897<\lambda$.

3) Use: $\lambda>\mu,\mu$ strong limit $\aleph_{0}<\kappa=\text{
cf}(\kappa)<\mu$ and for some $\theta\in\text{ Reg }\cap\mu\delimiter
2548495\kappa^{+}$, the pair $(Q,\kappa)$ fails the strong nonreflection.

We may change Definitions 1.4, 1.5 by the following.

We certainly gained though no real difference in the theorems in $\mathchar
632\relax$0.

###### Definition 4.7 Definition

1) We say that $\mathbb{F}$ is a $(\mathaccent 28694{C},\mathchar
7\relax,\theta)_{*}$-colouring

if:

2) If $\mathchar 7\relax=2$ we may omit it.

3) We may replace $\mathbb{F}$ by $\mathaccent 28694{F}=\langle
F_{\delta}:\delta\in S\rangle$ where $F_{\delta}$ is a function from
$\delimiter 69624584f:f$ a function from $[C_{\delta}]^{<\aleph_{0}}$ to
$\mathchar 7\relax\delimiter 86405897$ to $\theta$.

Abusing our notation, we do not strictly distinguish $\mathbb{F}$ and $F$:

4) Assume $\mathbb{F}$ is a $(\mathaccent 28694{C},\mathchar
7\relax,\theta)$-colouring, where $\mathaccent 28694{C}$ is a $\lambda$-Md-
parameter.

We say $\mathaccent 28694{c}\in{}^{S}\mathchar 7\relax$ (or $\mathaccent
28694{c}\in{}^{\lambda}\mathchar 7\relax$) is an $\mathbb{F}$-middle diamond
sequence, or an $\mathbb{F}$-Md-sequence if:

We also may say $\mathaccent 28694{c}$ is an $(\mathbb{F},S)$-Md-sequence or
any $(\mathbb{F},\mathaccent 28694{C})$-Md-sequence.

5) We say $\mathaccent 28694{c}\in{}^{S}\mathchar 7\relax$ is a
$D-{\mathbb{F}}$-Md-sequence if $D$ is a filter on $\lambda$ to which $S$
belongs and

6) We say $\mathaccent 28694{C}$ has the $(D,\theta)$-Md-property if

then there is a $D-\mathbb{F}$-Md-sequence, see part (3A). 7) We say that
$\mathaccent 28694{C}$ has the $(D,\mathchar 7\relax,\theta)$-Md-property as
in (5A) only $\mathbb{F}$ is a $(\mathaccent 28694{C},\mathchar
7\relax,\theta)$-colouring. We may omit $D$ if $D$ is the club filter. We may
omit $\theta$ if $\theta=2$.

8) We say $S\subseteq\lambda$ has the $\theta$-Md-property mean that some
$\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$ has. Now we shall
note that not much is changed.

$*\qquad*\qquad*$

We note below a dichotomy for a regular cardinal $\theta$.

###### \stag{q.1} Claim

Assume

1) If $\lambda>\mu$,

then for some $n_{\lambda}$ for every $n\in(n_{\lambda},\omega)$

${\text{\tenrm max pcf\/}}\delimiter 69624584\theta_{\alpha}:\alpha\in
A\delimiter 86405897\leq\lambda$. 2) Assume $\circledast_{\theta,\mu}$ which
is as defined in (1) only $C_{\delta}$ is a club of $\delta$ and $A\in
C_{\delta}$ is stationary, so, of course, $\kappa>\aleph_{0}$ (i.e.
NPr$(\lambda,\lambda_{n+1},\lambda_{n},\theta)$). If $\lambda=2^{2^{\mu_{*}}}$
then $S^{\lambda}_{\kappa}$ has the $(\mathchar 7\relax,\theta)$-Md-property
for $\mathchar 7\relax,\theta<\mu$.

3) In ? assume instead (e),

###### Demonstration Proof

1) We use [ Sh:513, $\mathchar 632\relax$1], i.e. we should define
${\mathfrak{D}}_{n}$.

? Sh:513 ?

2) Follows (note that the choice of $\mathaccent 28694{C}$ is immaterial and
having a counterexample to $\lambda$ gives one for $\lambda^{\prime}$).

###### \stag{q.2} Claim

Assume $\kappa=\mskip 3.0mu{\text{\tenrm cf/}}(\kappa)>\aleph_{0}$ and
$\neg(\forall\theta>\kappa)(\exists\lambda\in\mskip 3.0mu{\text{\tenrm
Reg\/}}\mskip 3.0mu\delimiter 2548495\theta)({\text{\tenrm
NPr\/}}(\lambda,\theta,\kappa)$. For some $\theta^{*}$ for every
$\lambda=\mskip 3.0mu{\text{\tenrm cf\/}}(\lambda)>\theta$ for some
$h:\lambda\rightarrow\theta^{*}$, if $\delta\in S^{\lambda}_{\kappa}$ then $h$
is one to one on some club of $\delta$.

###### \stag{q.3} Claim

Assume $\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$ is a
$(\lambda,\kappa)$-Md-parameter, $S\subseteq\lambda=\mskip 3.0mu{\text{\tenrm
cf\/}}(2^{\mu}),S\subseteq S^{\lambda}_{\kappa}$ is stationary and
${\text{\tenrm otp\/}}(C_{\delta})=\kappa>\aleph_{0}$, for $\delta\in S$ and
the condition (a)-(d),(f) of ? holds. Let $h_{\delta}=\mskip
3.0mu{\text{\tenrm OP\/}}_{\kappa,C_{\delta}}$.

Let $\langle f^{\delta}_{\alpha}:i<\lambda\rangle$ list ${}^{\kappa}(2^{\mu})$
and assume that the conclusion of ? fails.

Then for any $\nu\in{}^{\lambda}(2^{\mu})$ for some
$\eta\in{}^{\lambda}(2^{\mu})$ and club $E$ of $\lambda$ we have: if
$\delta\in S\cap E$ then $\eta\circ h_{\theta}\notin\delimiter
69624584f^{\delta}_{\alpha}:\alpha<\nu(\delta)\delimiter 86405897$; moreover,
$i<\nu(\delta)\Rightarrow\eta\circ
h_{\delta}\neq_{J^{bd}_{\kappa}}f^{\delta}_{i}$.

##  $\mathchar 632\relax$4 Further upgrading

We try get somewhat better results concerning upgrading and strongly
inaccessibles. In fact we show

###### Demonstration 4.1 Conclusion

1) Let $\mu>\aleph_{0}$ be strong limit. For some $\kappa_{0}<\mu$ for every
regular not strongly inaccessible $\lambda\geq 2^{2^{\mu}}$, for every
$\kappa\in\text{ Reg }\cap\mu\delimiter 2548495\kappa^{+}_{0}$ except possibly
one, $S^{\lambda}_{\kappa}$ has the $(\mathchar 7\relax,\theta)$-Md-property
(moreover, we can partition it to $\lambda$ sets each with that property).

2) If $\lambda=\text{ cf}(\lambda)$ is not strong limit and $2^{\mu}<y$ then
for some $\kappa_{*}=\text{ cf}(\kappa_{*})\leq\mu$ we have: if
$\lambda_{*}\leq\mu$ is regular, uncountable $\mathaccent 28694{C}$ is a
$(\lambda_{*},\kappa,\kappa^{+})$-Md-parameter consisting of clubs,
$\kappa=\text{ cf}(\kappa)\neq\kappa_{*}$ and $\mathaccent 28694{C}$ having
the $(\mathchar 7\relax,\theta)$-Md-property and $S\subseteq
S^{\lambda}_{\kappa}$ is stationary, $\in I[\lambda]$,

then for some $S$ has the $(\mathchar 7\relax,\theta)$-Md-property (witnessed
by a sequence of clubs).

###### Demonstration Proof

Let $\lambda_{*}=\text{ cf}(2^{\mu})$ and let $\kappa_{0}<\mu$ be regular such
that $\kappa\in\text{ Reg }\cap\mu\delimiter
2548495\kappa_{0}\wedge\alpha<\lambda_{*}\Rightarrow|\alpha|^{<\kappa>_{\text{tr}}}<\lambda_{*}$
and let $\mathchar 7\relax,\theta<\mu$. Now let $\kappa\in\text{ Reg
}\cap\mu\delimiter 2548495\kappa^{+}_{0}$, so there is a stationary
$S^{\kappa}_{1}\subseteq S^{\lambda_{*}}_{\kappa}$ which belong to
$I[\lambda]$ and let $\mathaccent 28694{a}=\langle
a_{\alpha}:\alpha<\lambda\rangle$ be such that $a_{\alpha}\subseteq\alpha$,
otp$(a_{\alpha})\leq\kappa,\alpha\in
S^{\kappa}_{1}\Leftrightarrow\alpha=\mathop{sup}\displaylimits(a_{\alpha}),\beta\in
a_{\alpha}\Rightarrow a_{\beta}=a_{\alpha}\cap\beta$. For $\delta\in
S^{\kappa}_{1}$, let $C^{1,\kappa}_{\delta}=\delimiter
69624584\gamma:\gamma<\alpha$ is a limit ordinal of cofinality (exactly)
$\kappa_{0}$ and
$\gamma=\mathop{sup}\displaylimits(a_{\alpha}\cap\gamma)\delimiter
86405897,D_{1}$ = the club filter on $\kappa+\delimiter
69624584\delta<\kappa:\text{cf}(\delta)=\text{ cf}(\kappa_{0})\delimiter
86405897$.

Now $\langle C^{1,\kappa}_{\delta}:\delta\in S^{\kappa}_{1}\rangle$ is a good
$(\lambda_{1},\kappa,\kappa^{+})$-Md-parameter. So by ? it has the $(\mathchar
7\relax,\theta)$-MD-property. Let $\lambda_{2}=\text{
cf}(\lambda_{2})>2^{\lambda_{1}}$, so it is enough to show $(\mathaccent
28694{C}^{1,\kappa}_{\delta},D_{\kappa})$ is liftable to $\lambda_{2}$ so by ?
+ ?, the parallel to $\mu$ there is chosen below, so we have to check the
conditions, mainly clause $(d)^{-}$ of ?.

Let $\chi=\text{ Min}\delimiter 69624584\chi:2^{\chi}\geq\lambda_{3}\delimiter
86405897$.

Case 1: $\chi$ is strong limit, cf$(\chi)<\mu$ in $\chi$. The one exceptional
cofinality is cf$(\chi)$ and when $\kappa\neq\text{ cf}(\chi)$ and the
parallel to $\mu$ in ?, the hypothesis $(d)^{\prime}$ of ? holds by ?(2),
i.e., at least one of them.

Case 2: $\chi$ strong limit, cf$(\chi)\geq\mu$.

So $\chi=\chi^{\mu}<\lambda\leq 2^{\chi}$, so we again use $\chi$ (and ?(2)).

Case 3: $\chi$ not strong limit.

So we can find $\sigma<\chi$ such that $2^{\sigma}\geq\chi$ so
$2^{\sigma}<\lambda$ by the choice of $\chi$ hence $\sigma<\chi\leq
2^{\sigma}<\lambda<2^{\chi}\leq 2^{2^{\sigma}}$ and
$(2^{\sigma})^{\sigma}=2^{\sigma}$ hence $(2^{\sigma})^{\mu}=2^{\sigma}$, so
we are done as in Case 2? using $2^{\sigma}$.

2) Similarly. $\square_{4.1}$

###### Remark Remark

So if $\lambda=\text{ cf}(\lambda)>2^{\mu}$ then for some $\kappa=\text{
cf}(\kappa)\leq\mu$ and $\chi$ the triple $(\lambda,\chi,\kappa)$ satisfies
the demands hence the conclusion of ?(2) for $(\lambda,\mu,\kappa)$, in fact,
$\chi$ is not strong limit or $\chi^{\mu}=\chi$.

Below we shed some light also being able to “lift”, in particular for the
strongly inaccessible case.

###### Definition 4.2 Definition

1) For regular uncountable $\kappa<\lambda$ and normal filter $D$ on $\kappa$
let $I^{*}_{1}[\lambda,\kappa,D]$ be the family of $A\subseteq\lambda$ such
that:

> for some club $E$ of $\lambda$ for every increasing continuous sequence
> $\langle\alpha_{\varepsilon}:\varepsilon<\kappa\rangle$ of ordinals from $E$
> we have $\delimiter 69624584\varepsilon<\kappa:\alpha_{\varepsilon}\in
> A\delimiter 86405897=\emptyset$ mod $D$.

2) For $\ell\in\delimiter 696245842,3\delimiter 86405897$ and $\lambda$
regular uncountable, $I^{*}_{\ell}[\lambda]$ is the family of
$A\subseteq\lambda$ such that some $E$ and $\mathaccent
28694{\Cal{P}}=\langle{\Cal{P}}_{\alpha}:\alpha<\lambda\rangle$ witness it
which means:

3) $I^{*}_{\ell}[\lambda,\kappa,D]$ is defined as in part (2) but in clause
(iv) is

###### Demonstration 4.3 Observation

: Let $\lambda$ be regular uncountable.

1) $I^{*}_{\ell}[\lambda]$ for $\ell=2,3$ is a normal idea on $\lambda$.

2) $I^{*}_{3}[\lambda]\subseteq I^{*}_{2}[\lambda]$.

3) If $\lambda$ is a successor cardinal,

then $I_{2}[\lambda]=I_{3}[\lambda]$.

4) If $\kappa\in\text{ Reg }\cap\lambda\delimiter 2548495\aleph_{1}$ and $D$
is a filter on $\kappa$, then $I^{*}_{\ell}[\lambda,\kappa,D]$ is a normal
filter on $\lambda$ for $\ell=2,3$ and $I^{*}_{3}[\lambda,\kappa,D]\subseteq
I^{*}_{2}[\lambda,\kappa,D]$.

$*\qquad*\qquad*$

The following claim ? helps us to show, e.g.,

###### Demonstration 4.4 Conclusion

If $\mu_{1}<\mu_{2}$ are strong limit singular,
cf$(\mu_{2})\neq\mu^{+}_{1},\kappa=\text{
cf}(\kappa)<\mu^{+}_{1},S^{\mu^{+}_{1}}_{\kappa}\notin I[\mu^{+}_{1}]$, so
necessarily cf$(\mu_{1})<\kappa$,

then some (in fact quite many) $S\subseteq S^{\mu^{+}_{2}}_{\kappa}$ has
middle diamond if some $S^{\mu^{+}_{1}}_{\kappa}$ has (where it holds in many
cases).

###### Demonstration Proof

By ? below and ? + ?.

In detail we apply claim ? for $\lambda=\mu^{+}_{2},\theta=\mu^{+}_{1},\kappa$
as above, $D=\delimiter
69624584A\subseteq\mu^{+}_{1}:S^{\mu^{+}_{1}}_{\kappa}\delimiter 2548495A\in
I[\mu^{+}_{1}]\delimiter 86405897$ or any normal filter.

By [ Sh:108] there is $S^{*}_{\ell}\subseteq\mu^{+}_{\ell}$, (the “bad” set)
such that $I[\mu^{+}_{\ell}]=\delimiter
69624584A\subseteq\mu^{+}_{\ell}:A\delimiter 2548495S^{*}_{\ell}$ not
stationary$\delimiter 86405897$. Let
$S_{2}=S^{\mu^{+}_{2}}_{\mu^{+}_{1}}\delimiter 2548495S^{*}_{2}$. Clearly the
assumptions (a) - (e) of ?(1) hold (clause (d) because $\mu_{2}$ is strong
limit and $\mu^{+}_{1}\neq\text{ cf}(\mu_{2}))$.

? Sh:108 ?

Letting $S_{1}=S^{\mu^{+}_{2}}_{\kappa}\delimiter 2548495S^{*}_{2}$ by [
Sh:108] clause (f) of ? hence all its assumptions hold. As clause (A) of ? of
its conclusion fail, necessarily clause (B) of ? holds. Let
$\lambda_{1}=\mu^{+}_{1},\lambda_{2}=\mu^{+}_{2},D_{\lambda_{1}}=D$.

? Sh:108 ?

By ? + ? we get the desired conclusion and we are done provided that its
assumption holds. Now (a), (b), (c), (e) are easy. For clause (a) and also
clause (f) using $D^{\prime}_{8}$ being a copy of $D$. We are left with clause
$(d)^{-}$, which holds as (B) of ? holds. $\square_{?}$

###### Definition 4.5 Definition

1) Let $\mathaccent 28694{C}=\langle C^{1}_{\delta}:\delta\in S_{1}\rangle$ be
a $(\lambda_{1},\kappa,\chi)$-Md-parameter, $\mathaccent 28694{D}=\langle
D_{\delta}:\delta\in S_{1}\rangle,D_{\delta}$ a filter on $C^{1}_{\delta}$ and
$D^{*}$ is a filter on $\lambda_{1}$. We say $(\mathaccent
28694{C},\mathaccent 28694{D},D^{*})$ is liftable to $\lambda_{2}$

if:

2) Instead $\mathaccent 28694{D}$ we may write one $D^{1}$ if
$\langle\text{otp}(C^{1}_{\delta}):\delta\in S_{1}\rangle$ is constant and
$D_{\delta}\cong D^{1}$ for $\delta\in S_{1}$. Instead $\mathaccent 28694{C}$
we may write $S\subseteq\lambda_{1}$ meaning every $\mathaccent
28694{C}=\langle C_{\delta}:\delta\in S_{1}\rangle,C_{\delta}$ is a clug of
$\delta$ of order type cf$(\delta)$.

4.6 Discussion: Of course, if we like to guess not $M\restriction
C^{2}_{\delta}$ but $M\restriction B$ for some $B\in D_{\delta}$, then (d)-
suffice.

###### \stag{4.5} Claim

Assume

Then at least one of the following occurs

###### Demonstration Proof

Let $\langle a_{\alpha}:\alpha<\lambda\rangle$ witness $S_{1}\in I[\lambda]$,
so $a_{\alpha}\subseteq\alpha,[\beta\in a_{\alpha}\Rightarrow
a_{\beta}=a_{\alpha}\cap\beta]$, otp$(a_{\alpha})\leq\kappa,\alpha\in
S_{1}\Leftrightarrow\alpha=\mathop{sup}\displaylimits(a_{\alpha})$ (possibly
replacing $S_{1}$ by $S_{1}\cap E$ for some club $E$ of $\lambda$).

For each $\delta\in S_{2}$ we choose $\mathaccent
28694{\alpha}_{\delta}=\langle\alpha_{\delta,\varepsilon}:\varepsilon<\theta\rangle$
increasing continuous with limit $\delta$. Possibly shrinking $S_{2}$, without
loss of generality

$\delta\in S_{2}\Rightarrow\delimiter
69624584\varepsilon:\alpha_{\delta,\varepsilon}\in S_{1}\delimiter 86405897\in
D$. Let $b_{\delta}=\delimiter
69624584\varepsilon<\theta:\alpha_{\delta,\varepsilon}\in S_{1}$ hence
cf$(\varepsilon)=\kappa$ and 121212if we demand
$``a_{\alpha_{\delta,\varepsilon}}\delimiter 2548495\cup\delimiter
69624584a_{\alpha_{\delta,\zeta}}:\zeta<\varepsilon\and\alpha_{\delta,\zeta}\in
S_{1}\delimiter 86405897$ is unbounded in $\alpha_{\delta,\varepsilon}"$ the
proof does not change.
$a_{\alpha_{\delta,\varepsilon}}\nsubseteq\cup\delimiter
69624584a_{\alpha_{\delta,\zeta}}:\zeta<\varepsilon\and\alpha_{\delta,\zeta}\in
S_{1}\delimiter 86405897\delimiter 86405897$. Let $S^{-}_{2}=\delimiter
69624584\delta\in S_{1}:b_{\delta}\neq\emptyset$ mod $D\delimiter 86405897$.

Case 1: $S^{-}_{2}$ is a stationary subset of $\lambda$.

For each $\delta\in S^{-}_{2}$ for every $\varepsilon\in b_{\delta}$ clearly
we have $\alpha_{\delta,\varepsilon}\in S_{1}$ hence
$\alpha_{\delta,\varepsilon}=\mathop{sup}\displaylimits(a_{\alpha_{\delta,\varepsilon}})$
and by the last phrase in the definition of $b_{\delta}$ there is
$\xi_{\delta,\varepsilon}<\varepsilon$ such that

As $D$ is normal (or just weakly normal), for some $\xi_{\delta}<\theta$

As $S^{-}_{2}$ is stationary and $2^{\theta}<\lambda$, for some stationary
$S^{-}_{3}\subseteq S^{-}_{2}$ and pair $(b^{*},\alpha^{*})$ we have

Let ${\Cal{P}}\subseteq[\alpha^{*}]^{\theta}$ exemplify
$\mathbb{U}_{D+b^{*}}(|\alpha^{*}|)<\lambda$, that is,

As $2^{\theta}<\lambda$, using a pairing function clearly there is ${\Cal{F}}$
such that

For each $\delta\in S^{-}_{3}$ we define a function
$f_{\delta}:b^{*}\rightarrow\alpha^{*}$ by $f_{\delta}(\varepsilon)=\text{
Min}(a_{\alpha_{\delta,\varepsilon}}\cap\alpha^{*}\delimiter
2548495\cup\delimiter
69624584a_{\alpha_{\delta,\zeta}}:\zeta<\varepsilon\and\zeta\in
b^{*}\delimiter 86405897)$, it is well defined as $b^{*}=b^{-}_{\delta}$
because $\delta\in S^{-}_{2}$. Clearly

We define $g_{\delta}:b^{*}\rightarrow\kappa$ by
$g_{\delta}(\varepsilon)=\text{ otp}(a_{\alpha_{\delta,\varepsilon}}\cap
f_{\delta}(\varepsilon))$ so for some
$i_{\delta}<\kappa,b^{2}_{\delta}=\delimiter 69624584\varepsilon\in
b^{*}:g(\varepsilon)=i\delimiter 86405897\in D^{+}$. Now for each $\delta\in
S^{-}_{3}$ by the choice of ${\Cal{F}}$ there is $b^{3}_{\delta}\subseteq
b^{2}_{\delta},b^{3}_{\delta}\neq\emptyset$ mod $D$ such that
$f_{\delta}\restriction b^{3}_{\delta}\in{\Cal{F}}$. As
$|{\Cal{F}}|=\mathbb{U}_{D+b^{*}}(|\alpha^{*}|)+2^{\theta}<\lambda$, for some
stationary $S^{-}_{4}\subseteq S^{-}_{3}$ and $(b^{**},i(*),f^{*})$ we have

By $\circledast_{4}$ we have

Now let $F:S_{1}\rightarrow\theta$ be such that $F(\alpha)=\varepsilon$ if
$f^{*}(\varepsilon)$ is equal to the $i(*)$-th member of $a_{\alpha}$. It
exemplifies clause $(B)$, one of the two possible conclusions of ?.

Case 2: $S_{2}\delimiter 2548495S^{-}_{2}$ stationary 131313but recall that we
have shrink $S_{2}$ in the beginning of the proof.

So we can choose $\delta\in S_{2}\delimiter 2548495S^{-}_{2}$ so
$b_{\delta}=\emptyset$ mod $D$. We choose by induction on $\varepsilon<\theta$
a pair $(h_{\varepsilon},\gamma_{\xi})$ such that:

There is no problem to carry the induction.

Let $h=\cup\delimiter 69624584h_{\varepsilon}:\varepsilon<\theta\delimiter
86405897$.

Let $E=\delimiter
69624584\varepsilon<\theta:\gamma_{\varepsilon}=\varepsilon\delimiter
86405897$, this is a club of $\theta$. Let $A=A_{\delta}=\delimiter
69624584\varepsilon:\varepsilon\in E\delimiter 2548495b_{\delta}$ and
$\alpha_{\varepsilon}\in S_{1}\delimiter 86405897$, so $A_{\delta}\in D$.

For $\varepsilon\in A_{\delta}$ we define
$C^{1}_{\varepsilon}=C^{1}_{\delta,\varepsilon}=\delimiter
69624584h(\beta):\beta\in a_{\alpha_{\delta,\varepsilon}}\delimiter 86405897$.
On $\theta$ we define a two-place relation $<^{*}$:

$<^{*}_{\delta}=\delimiter
69624584(h_{\varepsilon}(\beta_{1}),h_{\varepsilon}(\beta_{2})):\beta_{1}\in
a_{\beta_{2}}\text{ so necessarily }\beta_{2}\in S_{\theta}\delimiter
86405897.$

So $(\theta,<^{*}_{0})$ is a tree with $\kappa$ levels, and
$\alpha<^{*}\beta\Rightarrow\alpha<\beta$ (see clause (e)) and $\langle
C^{1}_{\delta,\varepsilon}:\varepsilon\in A\rangle$ is as required in clause
(A) of ? so we are done. $\square_{?}$

###### Remark 4.7 Remark

1) Note that we can apply ? varying $D$. We certainly can replace it by $D+A$,
hence we can strengthen (B) of the conclusion to:

2) If clause (d) of the assumption of ? holds for every normal filter
$D^{\prime}$ extending $D$,

then for some $E\in D$ and regressive $h:E\rightarrow\theta$, each
$A=h^{-1}\delimiter 69624584\alpha\delimiter 86405897$ is as required in
clause (B).

3) Instead of clause (c) of the assumption of ? we can assume

4) We can also use $D$ not a filter, just what is used on $D^{+}$.

5) We can in ? fix $\mathaccent 28694{a}$ and claim that (A) holds if Case 2
applies, possibility (B) holds if Case 1 applies.

###### Demonstration 4.8 Conclusion

Assume that

Then $S^{\lambda}_{\theta}$ has the $(\mathchar 7\relax,\theta)$-property
(using some sequence of clubs).

###### Demonstration Proof

As $S^{\lambda}_{\kappa}\in I[\kappa]$.

##  $\mathchar 632\relax$5 Relative of $I[\lambda]$

###### Definition 5.1 Definition

1) Assume that $\kappa<\lambda=\text{ cf}(\lambda)$ and $A\subseteq\kappa$ we
say that $\lambda$ weakly accept $(\kappa,A)$

if there is $\mathaccent
28694{\Cal{P}}=\langle{\Cal{P}}_{\alpha}:\alpha<\lambda\rangle$ witnessing it
which means

2) Assume $\kappa<\lambda=\text{ cf}(\lambda)$ and $D$ is a filter on
$\kappa$. We say that $\lambda$ weakly accepts $(\kappa,D)$ if there is
$\mathaccent 28694{\Cal{P}}=\langle{\Cal{P}}_{\alpha}:\alpha<\delta\rangle$
witnessing it which means:

3) Replacing $D$ by ${\Cal{P}}^{\prime}\subseteq{\Cal{P}}(\kappa)$ has the
same meaning and if $\emptyset\in{\Cal{P}}^{\prime}\subseteq{\Cal{P}}(\kappa)$
we mean ${\Cal{P}}(\kappa)\delimiter 2548495{\Cal{P}}^{\prime}$.

4) We say “strongly accept” if in the witness $\mathaccent 28694{\Cal{P}}$,
each ${\Cal{P}}_{\alpha}$ is a singleton.

5) We say “accept ” if in the witness $\mathaccent 28694{\Cal{P}}$ we have
$|{\Cal{P}}_{\alpha}|\leq|\alpha|$ (may say medium accept).

###### Definition 5.2 Definition

Let $\lambda$ be regular uncountable.

1) Let $I^{\prime}[\lambda]=I^{\prime}_{md}[\lambda]$ or
$I^{\prime}_{st}[\lambda]$ or $I^{\prime}_{wk}[\lambda]$ be the family of
$S\subseteq\lambda$ with a witness $\mathaccent 28694{\Cal{P}}$ which means
$\mathaccent
28694{\Cal{P}}=\langle{\Cal{P}}_{\alpha}:\alpha<\lambda\rangle,{\Cal{P}}_{\alpha}\subseteq{\Cal{P}}(\alpha)$
has cardinality $<\lambda$ or has cardinality $\leq 1$ or has cardinality
$\leq|\alpha|$ such that: for some club $E$ of $\lambda$

2) Assume $\lambda>\delta(*)$ and cf$(\delta(*))=\kappa$. For $x\in\delimiter
69624584md,st,wk\delimiter 86405897$ we define
$I^{\prime}_{x}[\lambda,\delta(*),\kappa]$ similarly only

then there is an unbounded $C\subseteq\delta$ satisfying otp$(C)=\delta(*)$
such that $\alpha\in C\cup\delimiter 69624584\delta\delimiter
86405897\Rightarrow C\cap\alpha\in{\Cal{P}}_{\alpha}$.

3) If $\lambda>\delta(*)=\mathop{sup}\displaylimits(A)\supseteq A$,
cf$(\delta(*))=\kappa$ we define $I^{\prime}[\lambda,A,\kappa]$ similarly only
in $(*)^{\prime}$ $C\in{\Cal{P}}_{\delta}$ and $\delimiter
69624584\text{otp}(\alpha\cap C):\alpha\in
C,C\cap\alpha\in{\Cal{P}}_{\alpha}\delimiter 86405897\supseteq A$.

4) Also for $D\subseteq{\Cal{P}}(\delta(*)),\emptyset\in D$ we can define
$I^{\prime}[\lambda,D,\kappa]$.

5.3 Fact: 1) In $(*)$ of the Definition 5.2(1), the demand otp$(C)<\delta$
gives an equivalent definition (i.e. by closing ${\Cal{P}}_{\alpha}$ as in [
Sh:420, $\mathchar 632\relax$1]).

? Sh:420 ?

2) $I^{\prime}[\lambda]$ is a normal ideal on $\lambda,\subseteq I[\lambda]$.

3) If $\lambda$ is a successor of regular cardinals then $\delimiter
69624584\delta<\lambda:\text{cf}(\delta)<\lambda\delimiter 86405897\in
I^{\prime}[\lambda]$.

###### Demonstration 5.4 Observation

If $\lambda>\kappa$ are regular then $\lambda$ accepts $(\mu,\kappa)$.

###### \stag{5.3} Claim

Assume

then $\delimiter 69624584a\in[\mu]^{\kappa}:{\text{\tenrm max
pcf\/}}\delimiter 69624584\lambda_{i}:i\in a\delimiter
86405897>\lambda\delimiter 86405897$ belongs to $D^{*}$.

Then we can find $\mathaccent
28694{\Cal{F}}=\langle{\Cal{F}}_{\alpha}:\alpha\in
S^{\lambda}_{\kappa}\rangle$ such that

then for some $\delta\in E,{\text{\tenrm cf\/}}(\delta)=\kappa$

and $f\in{\Cal{F}}_{\alpha}$ we have $f\subseteq g$ (can say “many”).

###### Demonstration Proof

Let $\langle{\Cal{P}}_{\alpha}:\alpha<\lambda\rangle$ witness that $\lambda$
accept $(\mu,D)$.

Let pr:$\lambda\times\lambda\rightarrow\lambda$ be one to one and let

$\displaystyle{\Cal{F}}^{*}_{\alpha}=\delimiter 69624584f:$ $\displaystyle
f\text{ is a partial function from }\alpha\text{ to }\alpha$
$\displaystyle\text{ with domain an unbounded subset of }\alpha\text{ of order
type }\kappa\delimiter 86405897$

${\Cal{F}}_{\alpha}=\delimiter 69624584f\in{\Cal{F}}^{*}_{\alpha}:\text{ the
set }\delimiter 69624584pr(\beta,f(\beta)):\beta\in\text{ Dom}(f)\delimiter
86405897\text{ belongs to }{\Cal{P}}_{\alpha}\delimiter 86405897.$

Given $g\in{}^{\lambda}\lambda$ let $E=\delimiter 69624584\delta:\text{if
}\alpha<\delta$ then sup$\delimiter 69624584f(\beta):\beta\leq\alpha\delimiter
86405897<\delta\delimiter 86405897$.

$Ag=\delimiter 69624584pr(\beta,g(\beta)):\beta<\lambda\delimiter 86405897$
and apply the definition 5.1.

Discussion: Now from the medium case, for cofinalities $<\beth_{\omega}$ get a
conclusion concerning the few-branches-property.

##  $\mathchar 632\relax$6 Relatives of $\clubsuit$

###### \stag{6v.1} Claim

Assume

Then there is $\mathaccent 28694{A}=\langle A^{*}_{\delta}:\delta\in S\rangle$
such that:

###### Demonstration Proof

Let $\mathaccent 28694{C}=\langle C_{\delta}:\delta\in S\rangle$ has the
$\kappa$-middle diamond.

We define a function $\mathbb{F}$, more exactly $\mathaccent
28694{\mathbb{F}}=\langle\mathbb{F}_{\delta}:\delta\in S\rangle$

$\text{Dom}(\mathbb{F}_{\alpha})=\delimiter 69624584f:f\text{ a two-place
function from }C_{\delta}\text{ to }\kappa\delimiter 86405897$

$\mathbb{F}_{\alpha}(f)=\text{ Min}\delimiter 69624584i:(\forall\alpha\in
C_{\alpha})(\exists\beta)(\alpha<\beta\in C_{\alpha})[f(\alpha,\beta)\leq
i]\delimiter 86405897.$

Note that $\mathbb{F}_{\alpha}(f)$ is well defined as $\kappa\neq\theta$. So
by assumption (d) we can find $\langle i^{*}_{\delta}:\delta\in S\rangle$ such
that

Let $\langle\lambda_{i}:i<\kappa\rangle$ be an increasing continuous sequence
of cardinals $<\mu$ with limit $\mu,\lambda_{0}>\kappa+0$.

For each $\alpha<\lambda$ let $\langle B_{\alpha,i}:i<\kappa\rangle$ be an
increasing continuous sequence of subsets of $\alpha$ with union $\alpha$ such
that $|B_{\alpha,i}|\leq\lambda_{i}$. At last we define $\mathaccent
28694{A}=\langle A_{\delta}:\delta\in S\rangle$ by

$A_{\delta}=\delimiter 69624584A_{\alpha,i^{*}_{\delta}}:\alpha\in
C_{\delta}\delimiter 86405897.$

Let us prove that $\mathaccent 28694{A}$ is as required so we are given
$A\subseteq\lambda=\mathop{sup}\displaylimits(A)$. We define a function
$f_{A}:\lambda\times\lambda\rightarrow\kappa$ by $\alpha<\beta<\lambda$ we let
$f_{A}(\alpha,\beta)=\text{ Min}\delimiter
69624584i:A\cap(\alpha,\beta)\neq\emptyset\delimiter 86405897$ and let
$E=\delimiter 69624584\delta<\lambda:\delta$ a limit ordinal and
$\delta=\mathop{sup}\displaylimits(A\cap\delta)\delimiter 86405897$ clearly a
club of $\lambda$. Now if $\delta\in S\cap E$ and
$i^{*}_{\delta}=\mathbb{F}_{\delta}(f_{A}\restriction C_{\delta})$ then
$\delta=\mathop{sup}\displaylimits(A_{\delta}\cap A)$; so clearly we are done.

###### Remark 6.1 Remark

Unfortunately, we do not know to find such $\mathaccent 28694{A}$ with
sup$\delimiter 69624584|A_{\delta}|:\delta\in S\delimiter 86405897<\mu$; see [
DjSh:xxx] why consistently at least for some such $\lambda$ this is inherent.
Maybe we can use squared middle diamond for each $\lambda_{i}$, seem doubtful.

? DjSh:xxx ?

###### \stag{6v.2} Claim

Assume

Then $\clubsuit_{S}$.

###### Demonstration Proof

We use a coding to $(2^{\mu_{*}})^{+}$-tuples, not all tuples can be coded so
we allow. So by this we consider ${\Cal{P}}_{\delta}$ as a $\delimiter
69624584\langle a_{\delta,j,i}:i<(2^{\mu_{*}})^{+}\rangle:j<\mu_{*}\delimiter
86405897$.

Each $i$ given an equivalence relation $E_{\delta,i}$ on $i$, one $X$ appear
$(2^{\mu_{*}})^{+}$. Use $*$ outside $X$ in $X$.

##  $\mathchar 632\relax$7 Successor of singular strong limit and being
Jonsson

This is a try

###### \stag{6.1} Claim

Assume $\kappa={\text{\tenrm
cf\/}}(\mu)<\mu,\kappa\neq\theta_{1}={\text{\tenrm
cf\/}}(\theta_{1})<\mu,\mu>{\text{\tenrm
cf\/}}(\theta_{2})=\theta_{2}\rightarrow(\theta_{1})^{2}_{\kappa}$ and
$S^{*}\subseteq S^{*}_{\theta_{2}},S^{*}\in I[\lambda]$ and

###### Demonstration Proof

See x.x (or earlier better claim using a scale $\mathaccent 28694{f}$ which
obeys $A$, so $\theta_{2}$ disappears?)

###### \stag{6.2} Claim

Assume

Then we can find $\langle b_{i}:i<\kappa\rangle$ with union $\delimiter
69624584\delta<\theta:{\text{\tenrm cf\/}}(\delta)\neq\kappa\delimiter
86405897$ and for each $i<\theta$ a partial square $\langle
a^{i}_{\alpha}:\alpha<\lambda\rangle$

###### Demonstration Proof

Similarly.

###### \stag{6.3} Claim

Assume

A try: Let $S_{1}\subseteq S^{\lambda}_{\kappa}$ be a stationary subset
reflecting in $S^{\lambda}_{\theta},\mathaccent 28694{C}^{1}=\langle
C^{1}_{\delta}:\delta\in S\rangle$ strict club system
(otp$(C_{\delta})=\kappa,\langle\text{cf}(\alpha):\alpha\in
C_{\delta}\rangle$) is increasing with limit
$\kappa<\theta=\text{cf}(\theta)<\mu$. But can we demand $\mathaccent
28694{C}$ is tree like?

Case 1: For every club $E$ of $\lambda$ for some sequence
$\langle\alpha_{\varepsilon}:\varepsilon<\theta\rangle$ increasing continuous,
$\alpha_{\varepsilon}\in E$ and the following set is stationary

$c_{\mathaccent 28694{\alpha}}=\bigl{\delimiter
69624584}\varepsilon<\theta:\alpha_{\varepsilon}\in S\wedge
C_{\alpha_{\varepsilon}}\subseteq^{*}E\wedge
c_{\alpha_{\varepsilon}}\nsubseteq^{*}\cup\delimiter
69624584c_{\alpha_{\zeta}}:\zeta\in\varepsilon\cap S\delimiter
86405897\bigr{\delimiter 86405897}.$

Then $\lambda\rightarrow[\lambda]^{<\omega}_{\theta}$ (reasonably also
$\lambda>[\lambda]^{2}_{\theta}$) as in [ Sh:g].

Case 2: Not Case 1 and $\mathaccent 28694{c}$ is tree like.

? Sh:g ?

###### Definition 7.1 Definition

Let $\mu,\sigma,\theta$ be cardinals, $C$ a set of ordinals and ${\Cal{A}}$ a
family of subsets of $C$.

1) For any set ${\Cal{S}}\subseteq\delimiter 69624584\mathaccent
28694{\eta}:\mathaccent
28694{\eta}=\langle\eta_{\varepsilon}:\varepsilon<\mu\rangle$ and
$\eta_{\varepsilon}\in{}^{c}\sigma\delimiter 86405897$ and
${\Cal{F}}\subseteq\delimiter 69624584\mathbb{F}:\mathbb{F}$ is a function
from $c_{\sigma}$ to $\theta\delimiter 86405897$.

We define a game $\Game=\Game(\mu,\sigma,\theta,{\Cal{S}},{\Cal{F}})$ between
Md and unif. A play last $\mu$ moves, in the $\varepsilon$-th move first the
Md-player chooses $\mathbb{F}_{\varepsilon}\in{\Cal{F}}$ and
$j_{\varepsilon}<\theta$ the unif player chooses
$\eta_{\varepsilon}\in{}^{c}\sigma$.

In the end the Md-player wins if
$\langle\eta_{\varepsilon}:\varepsilon<\theta\rangle\notin{\Cal{S}}$ or
$(\exists\varepsilon<\mu)(j_{\varepsilon}=\mathbb{F}_{\varepsilon}(\eta_{\varepsilon}))$.

2) Assume ${\Cal{A}}\subseteq{\Cal{P}}(C),{\Cal{S}}\subseteq\delimiter
69624584\mathaccent 28694{\eta}:\mathaccent
28694{\eta}=\langle\eta_{\varepsilon}:\varepsilon<\mu\rangle$ and each
$\eta_{\varepsilon}$ is a partial function from $C$ to $\sigma$,
Dom$(\eta_{\varepsilon})\in{\Cal{A}}\delimiter
86405897,\mathbb{F}\subseteq\delimiter 69624584\mathbb{F}:\mathbb{F}$ a
function from ${}^{A}\sigma$ to $\theta$ for some $A\in{\Cal{A}}\delimiter
86405897$ the game $\Game(\mu,\sigma,\theta,{\Cal{A}},{\Cal{S}},{\Cal{F}})$ is
defined similarly.

A play last $\mu$ moves, in the $\varepsilon$-th move first the MD-player
chooses $A_{\varepsilon}\in{\Cal{A}},\mathaccent
28694{\mathbb{F}}_{\varepsilon}\in{\Cal{F}}$ with domain
${}^{(A_{\varepsilon})}\sigma$ and $j_{\varepsilon}$ and second the unif-
player chooses $\eta_{\varepsilon}\in{}^{(A_{\varepsilon})}\sigma$. In the end
of the play the Md-player wins if
$\langle\eta_{\varepsilon}:\varepsilon<\mu\rangle\notin{\Cal{S}}$ or
$(\exists\varepsilon)(\mathbb{F}_{\varepsilon}(\eta_{\varepsilon})=j_{\varepsilon})$.

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[Sh 420]Saharon Shelah. Advances in Cardinal Arithmetic. In Finite and
Infinite Combinatorics in Sets and Logic, pages 355–383. Kluwer Academic
Publishers, 1993. N.W. Sauer et al (eds.).

[Sh:g]Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides.
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[Sh 430]Saharon Shelah. Further cardinal arithmetic. Israel Journal of
Mathematics, 95:61–114, 1996. math.LO/9610226.

[Sh 462]Saharon Shelah. On $\sigma$-entangled linear orders. Fundamenta
Mathematicae, 153:199–275, 1997. math.LO/9609216.

[Sh:f]Saharon Shelah. Proper and improper forcing. Perspectives in
Mathematical Logic. Springer, 1998.

[Sh 589]Saharon Shelah. Applications of PCF theory. Journal of Symbolic Logic,
65:1624–1674, 2000.

[Sh 460]Saharon Shelah. The Generalized Continuum Hypothesis revisited. Israel
Journal of Mathematics, 116:285–321, 2000. math.LO/9809200.

[Sh 576]Saharon Shelah. Categoricity of an abstract elementary class in two
successive cardinals. Israel Journal of Mathematics, 126:29–128, 2001.
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