# An almost strong relation

Shimon Garti  Institute of Mathematics, The Hebrew University of Jerusalem,
Jerusalem 91904, Israel<EMAIL_ADDRESS>and  Andrés Villaveces
Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá 111321,
Colombia<EMAIL_ADDRESS>
###### Abstract.

Let $\mu$ be a strong limit singular cardinal. We prove that if
$2^{\mu}>\mu^{+}$ then
$\binom{\mu^{+}}{\mu}\rightarrow\binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ for every
ordinal $\tau<\mu^{+}$. We obtain an optimal positive relation under
$2^{\mu}=\mu^{+}$, as after collapsing $2^{\mu}$ to $\mu^{+}$ this positive
relation is preserved.

###### Key words and phrases:

Polarized partition relations, elementary submodels, pcf theory

###### 2020 Mathematics Subject Classification:

03C55,03E04, 03E02, 03E05, 03E10, 05A18

##  0\. Introduction

The _strong polarized partition relation_
$\binom{\lambda}{\kappa}\rightarrow\binom{\lambda}{\kappa}_{\theta}$ says that
for every coloring $c:\lambda\times\kappa\rightarrow\theta$ one can find
$A\in[\lambda]^{\lambda},B\in[\kappa]^{\kappa}$ such that
$c^{\prime\prime}(A\times{B})$ is constant. The _almost strong polarized
partition relation_
$\binom{\lambda}{\kappa}\rightarrow\binom{\tau}{\kappa}_{\theta}$ for every
$\tau<\lambda$ asserts that for every coloring
$c:\lambda\times\kappa\rightarrow\theta$ and any ordinal $\tau<\lambda$ one
can find a color $i<\theta$ and a pair of sets
$A\subseteq\lambda,B\subseteq\kappa$ such that ${\rm otp}(A)=\tau,|B|=\kappa$
and $c^{\prime\prime}(A\times B)=\\{i\\}$. The purpose of this paper is to
prove such a relation at strong limit singular cardinals. The main theorem of
the paper improves both a result of Erdős-Hajnal-Rado for strong limit
singular cardinals with countable cofinality and a result of Shelah for strong
limit singular cardinals with uncountable cofinality.

Erdős, Hajnal and Rado showed in Theorem 42 of [EHR65] that whenever ${\rm
cf}(\mu)=\omega$, $\binom{\mu^{+}}{\mu}\rightarrow\binom{\mu}{\mu}_{2}$.
Assuming that $2^{\mu}>\mu^{+}$ and $\mu$ is a strong limit cardinal with
countable cofinality, we increase the order type of the first coordinate,
namely ${\rm otp}(A)=\tau$ for every ordinal $\tau<\mu^{+}$. Similarly, Shelah
proved in [She98] that
$\binom{\mu^{+}}{\mu}\rightarrow\binom{\mu+1}{\mu}_{<{\rm cf}(\mu)}$ whenever
$\mu>{\rm cf}(\mu)>\omega$, $\mu$ is strong limit and $2^{\mu}>\mu^{+}$.
Actually, Shelah’s theorem applies to singular cardinals of countable
cofinality as well, but in these cardinals the partition relation is less
interesting since it holds even if without the assumption that
$2^{\mu}>\mu^{+}$. Anyway, we show that the ordinal $\mu+1$ can be replaced by
every ordinal $\tau<\mu^{+}$ for singular cardinals of uncountable cofinality
under the same assumptions of $2^{\mu}>\mu^{+}$. These results give a positive
answer to Question 4.8 from [GS16]. Our proof follows in the footsteps of
Shelah’s proof in [She98], but we add a new feature which enables us to
stretch monochromatic sets of size $\mu$ and to obtain sets of the same
cardinality but a larger order type.

It is worth noting that the strong relation
$\binom{\mu^{+}}{\mu}\rightarrow\binom{\mu^{+}}{\mu}_{2}$ is consistent for
singular cardinals (see [GS12], [GS16]), using several assumptions about the
structure of cardinal arithmetic. For this relation the assumption
$2^{\mu}>\mu^{+}$ is necessary since $2^{\mu}=\mu^{+}$ implies
$\binom{\mu^{+}}{\mu}\nrightarrow\binom{\mu^{+}}{\mu}_{2}$ as proved in
[EHR65], and usually the positive strong relation requires more than just
$2^{\mu}>\mu^{+}$. But for our result, it suffices to assume $2^{\mu}>\mu^{+}$
in order to obtain the almost strong relation at strong limit singular
cardinals. Moreover, although collapsing $2^{\mu}$ to $\mu^{+}$ destroys the
strong relation $\binom{\mu^{+}}{\mu}\rightarrow\binom{\mu^{+}}{\mu}_{2}$ it
still preserves the almost strong relation, as will be indicated at the end of
the paper.

The result of Shelah was introduced in an expository article of Menachem
Kojman, [Koj95], who simplified the proof. The presentation here owes a lot to
Kojman’s paper, and in particular we follow the notation of that paper. We
divide the rest of the paper into two sections. In the first one we discuss
the general concept of pcf arrays of elementary submodels. In the second
section we prove the combinatorial theorem. The main reason for this
separation is that pcf arrays seem to represent a general method. Shelah
remarked that tentatively this concept might lead to other mathematical
proofs, though no such one has been discovered so far.

Our notation is standard. When we use elementary submodels of some
$\mathcal{H}(\chi)$ we mean that $\chi$ is a sufficiently large regular
cardinal and this structure can be augmented by any finite number of
additional predicates (like a well-ordering of the elements of
$\mathcal{H}(\chi)$). If $\kappa={\rm cf}(\kappa)<\lambda$ then
$S^{\lambda}_{\kappa}=\\{\delta\in\lambda:{\rm cf}(\delta)=\kappa\\}$. We
shall use this notation in most cases when $\lambda$ is a regular cardinal, in
which case $S^{\lambda}_{\kappa}$ is a stationary subset of $\lambda$. An
ordinal $\eta$ is indecomposable iff it has the form $\omega^{\beta}$, where
this denotes ordinal exponentiation. Indecomposable ordinals behave like
cardinals in the sense that if $\alpha<\eta$ then the order type of
$\eta-\alpha$ is still $\eta$. The collection of all indecomposable ordinals
of some $\lambda={\rm cf}(\lambda)>\aleph_{0}$ is a club subset of $\lambda$.
If $E$ is a club subset of $\lambda$ then ${\rm acc}(E)$ is the set of
accumulation points of $E$, and it is a club of $\lambda$ as well.

For basic background in pcf theory we suggest [AM10] and [BM90]. For advanced
theorems, including the main tool used in this paper, we refer to [She94]. A
good background about polarized relations appears in [Wil77] and in [HL10]. We
are grateful to the referee of the paper for a careful reading of the paper,
and for urging us to elaborate with regard to the case of singular cardinals
with countable cofinality. We also thank Saharon Shelah for a very helpful
discussion concerning several issues in pcf theory.

##  1\. Pcf arrays of elementary submodels

We define in this section the notion of a _pcf array of elementary submodels_.
We provide sufficient conditions for the existence of this kind of arrays.
This will be useful in the proof of the almost strong polarized relation later
in the paper.

Let $\mu$ be a singular cardinal, $\kappa={\rm cf}(\mu)$. Let
$\langle\mu_{\alpha}:\alpha<\kappa\rangle$ be an increasing sequence of
regular cardinals such that $\mu=\bigcup_{\alpha<\kappa}\mu_{\alpha}$. Let $J$
be an ideal over $\kappa$ which contains the ideal $J^{\rm bd}_{\kappa}$ of
all bounded subsets of $\kappa$.

The relation $f<_{J}g\Leftrightarrow\\{\alpha\in\kappa:g(\alpha)\leq
f(\alpha)\\}\in J$ defined on elements from the product
$\prod\limits_{\alpha<\kappa}\mu_{\alpha}$ is usually just a partial order. As
such, it may, or may not, possess a $J$-increasing cofinal sequence of
functions. If there is a cofinal $J$-increasing sequence then the minimal
length of such a $J$-increasing sequence is denoted by ${\rm
tcf}(\prod\limits_{\alpha<\kappa}\mu_{\alpha},J)$ and called _the true
cofinality_ of the product.

If $J$ is a prime ideal (equivalently, if the dual of $J$ is an ultrafilter)
then the partial order defined above is actually a linear order and hence a
cofinal sequence exists. The spectrum of possible cofinalities is the most
basic concept of pcf theory. Formally, if $\mathfrak{a}$ is a set of regular
cardinals then ${\rm pcf}(\mathfrak{a})=\\{{\rm
tcf}(\prod\mathfrak{a},\mathscr{U}):\mathscr{U}$ is an ultrafilter over
$\mathfrak{a}\\}$. For the majority of pcf theorems, $\mathfrak{a}$ has to be
progressive, where a set of regular cardinals $\mathfrak{a}$ is called
_progressive_ iff $|\mathfrak{a}|<\min(\mathfrak{a})$. In many cases the set
$\mathfrak{a}$ is an end-segment of the set of all regular cardinals below a
given singular cardinal $\mu$. For this set to be progressive one has to
assume that $\mu$ is not a fixed point of the $\aleph$-function.

The elements of ${\rm pcf}(\mathfrak{a})$ can be characterized by ideals of
the form $J_{<\lambda}[\mathfrak{a}]$. To define these ideals, suppose that
$\mathfrak{a}$ is progressive and $\mathfrak{b}\subseteq\mathfrak{a}$. One
says that $\mathfrak{b}$ _dictates_ cofinality less than $\lambda$ if ${\rm
tcf}(\prod\mathfrak{a},\mathscr{U})<\lambda$ whenever $\mathscr{U}$ is an
ultrafilter over $\mathfrak{a}$ and $\mathfrak{b}\in\mathscr{U}$. The ideal
$J_{<\lambda}[\mathfrak{a}]$ is the collection of subsets of $\mathfrak{a}$
which dictate cofinality less than $\lambda$. By classical pcf theorems, if
$\lambda\in{\rm pcf}(\mathfrak{a})$ then
$J_{<\lambda^{+}}[\mathfrak{a}]=J_{<\lambda}[\mathfrak{a}]+\mathfrak{b}$ for
some $\mathfrak{b}\subseteq\mathfrak{a}$. Namely, the ideal
$J_{<\lambda^{+}}[\mathfrak{a}]$ is generated over the ideal
$J_{<\lambda}[\mathfrak{a}]$ by a single set. This property is called
_normality_ , and the set $\mathfrak{b}$ is called _a generator_. It is not
unique as a set, but if $\mathfrak{b}_{0},\mathfrak{b}_{1}$ both generate
$J_{<\lambda^{+}}[\mathfrak{a}]$ over $J_{<\lambda}[\mathfrak{a}]$ then the
symmetric difference $\mathfrak{b}_{0}\triangle\mathfrak{b}_{1}$ belongs to
$J_{<\lambda}[\mathfrak{a}]$.

A fundamental theorem of pcf theory is that if $\mu>{\rm cf}(\mu)=\kappa$ then
there exists a sequence $\langle\mu_{\alpha}:\alpha\in\kappa\rangle$ such that
${\rm tcf}(\prod_{\alpha\in\kappa}\mu_{\alpha},J^{\rm bd}_{\kappa})=\mu^{+}$.
According to this theorem we can realize $\mu^{+}$ as a true cofinality using
the ideal of bounded subsets of $\kappa$. In general, it might happen that
$\lambda\in{\rm pcf}(\mathfrak{a})$ where $\mathfrak{a}\subseteq{\rm
Reg}\cap\mu$ and $\lambda$ cannot be expressed as a true cofinality with the
ideal $J^{\rm bd}_{\kappa}$. But in the case of $\lambda=\mu^{++}$ we have the
following result from [She94, Chapter VIII, Theorem 1.1] at singular cardinals
of uncountable cofinality. Since the case of countable cofinality is not
explicit in this monograph, we unfold the argument. See page 2.2 of this paper
for a more detailed explanation. We also refer the reader to [Koj95] at this
point.

######  Theorem 1.1.

Assume that $\mu>{\rm cf}(\mu)=\kappa\geq\aleph_{0}$, and $\mu$ is a strong
limit cardinal. Assume further that $2^{\mu}>\mu^{+}$. Then there exists an
increasing sequence of regular cardinals $\langle\lambda_{i}:i<\kappa\rangle$
such that $\mu=\bigcup_{i<\kappa}\lambda_{i}$ and ${\rm
tcf}(\prod_{i<\kappa}\lambda_{i},J^{\rm bd}_{\kappa})=\mu^{++}$.

_Proof_.
Firstly, we argue that $\mu^{++}\leq pp(\mu)$ (there is some ideal $J$ on
$\kappa$ and some increasing sequence of regular cardinals
$\langle\lambda_{i}:i<\kappa\rangle$ such that ${\rm
tcf}(\prod_{i<\kappa}\lambda_{i},J)\geq\mu^{++}$). Then we shall prove that
this can be realized by the ideal $J^{\rm bd}_{\kappa}$. For the first task,
notice that $2^{\mu}=\mu^{\kappa}$ as $\mu$ is a strong limit singular
cardinal. For simplicity, we assume that $\mu$ is not a fixed point of the
$\aleph$-function.111The statement of the theorem holds at fixed points of the
aleph function as well, for details see [Koj95]. Hence there is an interval of
regular cardinals $\mathfrak{a}\subseteq\mu$ such that
$|\mathfrak{a}|^{+}<\min(\mathfrak{a})$ and even
$2^{|\mathfrak{a}|}<\min(\mathfrak{a})$. Under the above assumption, ${\rm
pcf}(\mathfrak{a})$ is an interval of regular cardinals as well. From [BM90,
Theorem 5.1] we know that $\max{\rm
pcf}(\mathfrak{a})=|\prod\mathfrak{a}|=\mu^{\kappa}=2^{\mu}$. Thus,
$pp(\mu)\geq\max{\rm pcf}(\mathfrak{a})\geq\mu^{++}$ as we are assuming that
$2^{\mu}>\mu^{+}$.

Now, since ${\rm pcf}(\mathfrak{a})$ is an interval of regular cardinals (this
is a consequence of the no-holes theorem, see [AM10, Theorem 3.1] or [BM90,
Corollary 2.2]) one concludes that $\mu^{++}\in{\rm pcf}(\mathfrak{a})$.
Specifically, for some increasing sequence
$\langle\lambda_{i}:i\in\kappa\rangle$ of regular cardinals so that
$\mu=\bigcup_{i\in\kappa}\lambda_{i}$ it is true that $\mu^{++}={\rm
tcf}(\prod_{i\in\kappa}\lambda_{i},J)$ where $J\supseteq J^{\rm bd}_{\kappa}$.

We move to the second task, that is, we explain why we can assume that the
ideal $J$ can be taken as $J^{\rm bd}_{\kappa}$ itself. For every
$\lambda\in{\rm pcf}(\mathfrak{a})$ we fix a generator
$\mathfrak{b}_{\lambda}$. By classical pcf theorems (see, e.g., [BM90,
Corollary 4.4]) one has $\lambda={\rm
tcf}(\prod\mathfrak{b}_{\lambda},J_{<\lambda}[\mathfrak{a}])$. In particular,
$\mu^{++}={\rm
tcf}(\prod\mathfrak{b}_{\mu^{++}},J_{<\mu^{++}}[\mathfrak{a}])$.

Let us focus on the two generators $\mathfrak{b}_{\mu^{+}}$ and
$\mathfrak{b}_{\mu^{++}}$. The crucial point is that we may assume that
$\mathfrak{b}_{\mu^{+}}\cap\mathfrak{b}_{\mu^{++}}=\varnothing$ upon replacing
$\mathfrak{b}_{\mu^{++}}$ by $\mathfrak{b}_{\mu^{++}}-\mathfrak{b}_{\mu^{+}}$,
see [AM10, Theorem 4.8] and [AM10, Notation 4.9]. Recall that $\mu$ is a
singular cardinal, and hence
$J_{<\mu^{+}}[\mathfrak{a}]=J_{<\mu}[\mathfrak{a}]$. But
$J_{<\mu}[\mathfrak{a}]$ is none other than the ideal $J^{\rm bd}_{\kappa}$,
since $\mu=\bigcup\mathfrak{a}$. Finally, since
$J_{<\mu^{++}}[\mathfrak{a}]=J_{<\mu^{+}}[\mathfrak{a}]+\mathfrak{b}_{\mu^{++}}$
and $\mathfrak{b}_{\mu^{+}}$ is disjoint from $\mathfrak{b}_{\mu^{++}}$, the
ideal $J_{<\mu^{++}}[\mathfrak{a}]$ equals the ideal
$J_{<\mu^{+}}[\mathfrak{a}]$ over the generator $\mathfrak{b}_{\mu^{++}}$.
Thus, after replacing $\mathfrak{b}_{\mu^{++}}$ by
$\mathfrak{b}_{\mu^{++}}-\mathfrak{b}_{\mu^{+}}$ one has
$J_{<\mu^{++}}[\mathfrak{a}]=J^{\rm bd}_{\kappa}$, and we are done.

$\qed_{\ref{thmjbd}}$

One of the virtues of $J^{\rm bd}_{\kappa}$ is that if we take any unbounded
subsequence $\langle\lambda_{i_{\beta}}:\beta\in\kappa\rangle$ of the above
sequence then ${\rm tcf}(\prod_{\beta\in\kappa}\lambda_{i_{\beta}},J^{\rm
bd}_{\kappa})=\mu^{++}$ as well. Indeed, if $(f_{\alpha}:\alpha\in\mu^{++})$
witnesses ${\rm tcf}(\prod_{i<\kappa}\lambda_{i},J^{\rm
bd}_{\kappa})=\mu^{++}$ then the restriction of the scale to the subsequence
$\langle\lambda_{i_{\beta}}:\beta\in\kappa\rangle$ witnesses ${\rm
tcf}(\prod_{\beta\in\kappa}\lambda_{i_{\beta}},J^{\rm bd}_{\kappa})=\mu^{++}$.

In general, it is possible that $\lambda={\rm
tcf}(\prod_{i<\kappa}\lambda_{i},J)$ for some $J\supsetneqq J^{\rm
bd}_{\kappa}$ and $\lambda$ cannot be realized as the true cofinality of some
sequence of regular cardinals below $\mu$ with the ideal $J^{\rm
bd}_{\kappa}$. In such cases we do not know how to carry out our argument. The
problematic point is that by thinning-out the sequence to
$\langle\lambda_{i_{\beta}}:\beta\in\kappa\rangle$ we may change the true
cofinality. However, in some cases one can ensure that the true cofinality is
obtained by the ideal $J^{\rm bd}_{\kappa}$. A notable example is when
$\mu>{\rm cf}(\mu)=\kappa>\aleph_{0}$ and $\kappa$ is weakly compact, as
proved in [GS13]. Another case, crucial for our proof, is the case in which
$\lambda=\mu^{++}$, see [She98, Theorem 1.4].

In what follows, we will define matrices of models $(M_{\alpha
i})_{\alpha<\lambda,i<\kappa}$, where $\mu>{\rm cf}(\mu)=\kappa$ and
$\lambda\geq\mu^{+}$; furthermore, for some increasing sequence of regular
cardinals $\langle\lambda_{i}:i<\kappa\rangle$,
$\mu=\bigcup_{i<\kappa}\lambda_{i}$. This last part connects the
$\lambda\times\kappa$-matrix to the statement $\lambda={\rm
tcf}(\prod_{i<\kappa}\lambda_{i},J)$.

######  Definition 1.2 (Pcf array).

Let $\mu$ be a singular cardinal, $\kappa={\rm cf}(\mu)$ and $\chi$ a
sufficiently large regular cardinal above $2^{\mu}$.
Assume that $\mu<\lambda<{\rm tcf}(\prod_{i<\kappa}\lambda_{i},J)$,
$J\supseteq J_{\kappa}^{\rm bd}$ and for some increasing sequence of regular
cardinals $\bar{\lambda}=\langle\lambda_{i}\mid i<\kappa\rangle$,
$\mu=\bigcup_{i<\kappa}\lambda_{i}$.
A $\lambda\times\kappa$ _pcf array of models_ $\mathcal{M}$ is a sequence of
elementary submodels of $\mathcal{H}(\chi)$ of the form $\langle M_{\alpha
i}:\alpha<\lambda,i<\kappa\rangle$ such that (see Figure 1):

  1. $(\aleph)$

For every $\alpha\in\lambda,M_{\alpha i}\prec M_{\alpha j}$ whenever $i<j$.

  2. $(\beth)$

For every $\alpha<\beta<\lambda$ there exists $u\in J$ such that $M_{\alpha
j}\prec M_{\beta j}$ whenever $j\in\kappa-u$.

  3. $(\gimel)$

$|M_{\alpha i}|<\lambda_{i}$ for every $\alpha\in\lambda,i\in\kappa$.

We shall say that $\mathcal{M}$ is _based on the sequence_ $\bar{\lambda}$
_and the ideal_ $J$. The following figure demonstrates the idea in the case
where $J=J^{\rm bd}_{\kappa}$; in this case, for every $\alpha<\beta<\lambda$
there exists an ordinal $i_{\alpha\beta}<\kappa$ such that $M_{\alpha j}\prec
M_{\beta j}$ whenever $i_{\alpha\beta}\leq j<\kappa$.

$\vdots$ $\vdots$ $\vdots$

$\ddots$

$M_{\beta 0}$ $\prec$ $M_{\beta i_{\alpha\beta}}$ $\prec$ $M_{\beta i}$
$\cdots$$\vdots$

$\prec$

$\prec$

$M_{\alpha 0}$ $\prec$ $M_{\alpha i_{\alpha\beta}}$ $\prec$ $M_{\alpha i}$
$\cdots$$\vdots$$\vdots$ $\vdots$ $\cdots$$M_{00}$ $\prec$
$M_{0i_{\alpha\beta}}$ $\prec$ $M_{0i}$ $\cdots$$\lambda$$\kappa$ Figure 1. A
pcf-array

A central concept related to pcf arrays is the following:

######  Definition 1.3 (The characteristic sequence).

Let $\mathcal{M}=\langle M_{\alpha i}:\alpha<\lambda,i<\kappa\rangle$ be a pcf
array of models based on $\bar{\lambda}=\langle\lambda_{i}:i<\kappa\rangle$.
For every $\alpha<\lambda,i<\kappa$ let $f_{\alpha}(i)=\sup(M_{\alpha
i}\cap\lambda_{i})$. Each function
$f_{\alpha}\in\prod\limits_{i<\kappa}\lambda_{i}$ is called the
_characteristic function_ of $\mathcal{M}$ at stage $\alpha$, and
$(f_{\alpha})_{\alpha<\lambda}$ is called the _characteristic sequence_.

Notice that each $f_{\alpha}$ is an element of
$\prod\limits_{i<\kappa}\lambda_{i}$ since $|M_{\alpha i}|<\lambda_{i}$. The
main point of the proof in the next section is that for a pcf array
$\mathcal{M}$ and $\lambda$-many functions $f_{\alpha}$ as above there is a
single function $h\in\prod\limits_{i<\kappa}\lambda_{i}$ which bounds them
all. The reason is that ${\rm
tcf}(\prod\limits_{i<\kappa}\lambda_{i},J)>\lambda$. Now the values of $h$
will be ordinals outside the pertinent models; later in the proof we will use
the fact that any first order property of them will be reflected to many
ordinals below.

######  Claim 1.4.

Let $\mu$ be a strong limit singular cardinal, $\kappa={\rm cf}(\mu)>\omega$
and $2^{\mu}>\mu^{+}$. Let $\bar{\mu}=\langle\mu_{i}:i<\kappa\rangle$ be an
increasing continuous sequence of cardinals such that
$\mu=\bigcup_{i<\kappa}\mu_{i}$ and $\mu_{0}>\kappa$. Fix also another
increasing sequence of cardinals
$\bar{\lambda}=\langle\lambda_{i}:i<\kappa\rangle$ such that
$\mu=\bigcup_{i<\kappa}\lambda_{i}$.
There exists a pcf array $\mathcal{M}=\langle M_{\alpha
i}:\alpha<\mu^{+},i<\kappa\rangle$ based on
$\bar{\lambda}=\langle\lambda_{i}:i<\kappa\rangle$ and $J^{\rm bd}_{\kappa}$
such that ${}^{\mu_{i}^{+}}M_{\alpha i}\subseteq M_{\alpha i}$ for every
$\alpha\in\mu^{+},i<\kappa$ and $\alpha\subseteq\bigcup_{i<\kappa}M_{\alpha
i}$ for every $\alpha\in\mu^{+}$.

_Proof_.
We may assume, without loss of generality, that
$2^{(\mu_{i}^{+})}<\lambda_{i}$ for every $i<\kappa$ by taking a subsequence
of $\bar{\lambda}$ if needed. We use here the fact that $\mu$ is a strong
limit cardinal. Choose for every $\alpha\in\mu^{+},i<\kappa$ some $M_{\alpha
i}\prec\mathcal{H}(\chi)$ such that:

  1. $(a)$

$|M_{\alpha i}|=2^{(\mu_{i}^{+})}$ and $2^{(\mu_{i}^{+})}\subseteq M_{\alpha
i}$.

  2. $(b)$

${}^{\mu_{i}^{+}}M_{\alpha i}\subseteq M_{\alpha i}$.

  3. $(c)$

$\alpha\in M_{\alpha i}$ and $\alpha\subseteq\bigcup_{i<\kappa}M_{\alpha i}$.

  4. $(d)$

$\\{M_{\alpha i}:(\alpha,i)<_{\rm lex}(\beta,j)\\}\in M_{\beta j}$.

The only non-trivial requirement in the construction is
$\alpha\subseteq\bigcup_{i<\kappa}M_{\alpha i}$, since typically $\mu<\alpha$
and $|M_{\alpha i}|<\mu$ for every $i\in\kappa$. However, $\alpha<\mu^{+}$ so
$\alpha$ is expressible as $\bigcup_{i<\kappa}\alpha_{i}$ such that
$|\alpha_{i}|\leq 2^{(\mu_{i}^{+})}$ for every $i\in\kappa$, and the
construction follows.

We now explain why the construction works. Requirement $(\gimel)$ of
Definition 1.2 is guaranteed by $(a)$ and our assumption on the
$\lambda_{i}$’s, so we check $(\aleph)$ and $(\beth)$. We first note that if
$(\alpha,i)<_{\rm lex}(\beta,j)$ and $\alpha\in M_{\beta j}$ then $M_{\alpha
i}\in M_{\beta j}$. This follows from $(d)$, and from the fact that $i\in
M_{\beta j}$ as $\kappa\in 2^{(\mu_{i}^{+})}\subseteq M_{\beta j}$. Second, if
$M_{\alpha i}\in M_{\beta j}$ and $j\leq i$ then $M_{\alpha i}\subseteq
M_{\beta j}$. Indeed, $\mathcal{H}(\chi)\models|M_{\alpha
i}|=2^{(\mu_{i}^{+})}$ so there is a function $g:2^{(\mu_{i}^{+})}\rightarrow
M_{\alpha i}$ which enumerates its elements. Since $M_{\beta
j}\prec\mathcal{H}(\chi)$ there is such a function in $M_{\beta j}$ and since
$2^{(\mu_{i}^{+})}\leq 2^{(\mu_{j}^{+})}\subseteq M_{\beta j}$ we see that the
range of this function is a subset of $M_{\beta j}$, namely $M_{\alpha
i}\subseteq M_{\beta j}$. Finally, if $M_{\alpha i}\subseteq M_{\beta j}$ then
$M_{\alpha i}\prec M_{\beta j}$ since both are elementary submodels of the
same $\mathcal{H}(\chi)$.

Fix any $\alpha\in\mu^{+}$. If $i<j$ then $(\alpha,i)<_{\rm lex}(\alpha,j)$.
Since $\alpha\in M_{\alpha j}$ by $(c)$, we infer that $M_{\alpha i}\prec
M_{\alpha j}$ so $(\aleph)$ holds. To check $(\beth)$, assume that
$\alpha<\beta$. Since $\beta\subseteq\bigcup_{i\in\kappa}M_{\beta i}$ and
$\alpha<\beta$ there is $i_{\alpha\beta}\in\kappa$ such that $\alpha\in
M_{\beta i_{\alpha\beta}}$. If $j\geq i_{\alpha\beta}$ then $M_{\beta
i_{\alpha\beta}}\subseteq M_{\beta j}$ and hence $\alpha\in M_{\beta j}$. As
$(\alpha,j)<_{\rm lex}(\beta,j)$ we have $M_{\alpha j}\prec M_{\beta j}$ thus
proving $(\beth)$.

$\qed_{\ref{clmexist}}$

##  2\. Almost strong relations

In this section we shall prove the combinatorial results concerning the
polarized relation. We first provide a useful lemma which says, roughly, that
if a coloring of pairs depends only on one coordinate then it has large
monochromatic products. This observation is pertinent for both cases of
countable and uncountable cofinality.

######  Lemma 2.1.

Assume that $\mu>{\rm cf}(\mu),\tau<\mu^{+}$ and $\theta<{\rm cf}(\mu)$.
Let $c:\mu^{+}\times\mu\rightarrow\theta$ be any coloring.
If $A\subseteq\mu^{+},{\rm otp}(A)=\tau,B_{0}\in[\mu]^{\mu}$ and
$c\upharpoonright(A\times B_{0})$ depends only on the second coordinate then
there is a set $B\in[B_{0}]^{\mu}$ such that $c\upharpoonright(A\times B)$ is
constant.

_Proof_.
Define a function $d:B_{0}\rightarrow\theta$ as follows. Fix any $\alpha\in A$
and let $d(\beta)=c(\alpha,\beta)$ for every $\beta\in B_{0}$. The choice of
$\alpha$ is unimportant by the assumption of the lemma. Since $|B_{0}|=\mu$
and $\theta<{\rm cf}(\mu)$ one can find $j\in\theta$ and $B\in[B_{0}]^{\mu}$
such that $\beta\in B\Rightarrow d(\beta)=j$. It follows that
$c(\alpha,\beta)=j$ for every $\alpha\in A,\beta\in B$, so we are done.

$\qed_{\ref{lem0}}$

The above lemma will be used within the proof of the main result of the paper,
which reads as follows.

######  Theorem 2.2.

Assume that $\mu>{\rm cf}(\mu)=\kappa\geq\aleph_{0}$, $\mu$ is a strong limit
cardinal and $2^{\mu}>\mu^{+}$. Then
$\binom{\mu^{+}}{\mu}\rightarrow\binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ for every
$\tau<\mu^{+}$.

_Proof_.
Fix two sequences of cardinals, $\bar{\mu}=\langle\mu_{i}:i<\kappa\rangle$ and
$\bar{\lambda}=\langle\lambda_{i}:i<\kappa\rangle$, with the following
properties:

  1. $(a)$

$\bar{\mu}$ is increasing and continuous, $\kappa<\mu_{0}$ and
$\mu=\bigcup_{i<\kappa}\mu_{i}$.

  2. $(b)$

$\bar{\lambda}$ is an increasing sequence of regular cardinals,
$\mu=\bigcup_{i<\kappa}\lambda_{i}$.

  3. $(c)$

For every $i<\kappa,2^{\mu_{i}^{+}}<\lambda_{i}$.

  4. $(d)$

${\rm tcf}(\prod_{i<\kappa}\lambda_{i},J^{\rm bd}_{\kappa})=\mu^{++}$.

Part $(d)$ is possible by Theorem 1.1. Part $(c)$ is possible since $\mu$ is a
strong limit cardinal and $\bar{\lambda}$ can be replaced by any unbounded
subsequence while keeping $(d)$. The fact that we use $J^{\rm bd}_{\kappa}$ in
the representation of $\mu^{++}$ at $(d)$ is crucial here.

Fix any $\theta<\kappa$ and a coloring $c:\mu^{+}\times\mu\rightarrow\theta$.
Let $\tau<\mu^{+}$ be an ordinal. By the monotonicity of the arrow notation we
may increase $\tau$ and therefore assume that ${\rm cf}(\tau)=\kappa$ and
$\tau=\bigcup_{i<\kappa}\delta_{i}$ where each $\delta_{i}$ is indecomposable
and $\delta_{i}>\bigcup_{j<i}\delta_{j}$ for every $i<\kappa$. We also choose
another sequence of indecomposable ordinals $\langle\nu_{i}:i\in\kappa\rangle$
so that $\delta_{i}<\nu_{i}<\delta_{i+1}$ for every $i\in\kappa$. These
assumptions are possible since $S^{\mu^{+}}_{\kappa}$ is stationary and the
set of indecomposable ordinals below $\mu^{+}$ is a club.

Let $\chi$ be a sufficiently large regular cardinal in which the arguments of
our proof will be carries out. Let $\mathcal{M}=\langle M_{\alpha
i}:\alpha<\mu^{+},i<\kappa\rangle$ be a pcf array of submodels of
$\mathcal{H}(\chi)$, based on the ideal $J^{\rm bd}_{\kappa}$ and the
sequences $\bar{\mu},\bar{\lambda}$ and let $\langle
f_{\alpha}:\alpha<\mu^{+}\rangle$ be its characteristic sequence. We may
assume that ${}^{\kappa}\theta\subseteq M_{\alpha i}$ for every
$\alpha\in\mu^{+},i\in\kappa$. Since ${\rm
tcf}(\prod_{i<\kappa}\lambda_{i},J^{\rm bd}_{\kappa})=\mu^{++}$, we can fix a
function $f^{*}\in\prod_{i<\kappa}\lambda_{i}$ such that
$\alpha\in\mu^{+}\Rightarrow f_{\alpha}<_{J^{\rm bd}_{\kappa}}f^{*}$.

We shall use $f^{*}$ as a _translation_ of the coloring $c$, defined on pairs,
to a function defined on singletons in the following sense: for every
$\alpha\in\mu^{+}$ define $g_{\alpha}\in{}^{\kappa}\theta$ by letting
$g_{\alpha}(i)=c(\alpha,f^{*}(i))$. Here we still depend on both coordinates,
but since $\theta^{\kappa}<\mu$ there is a fixed $g^{*}\in{}^{\kappa}\theta$
such that the set $H=\\{\alpha\in\mu^{+}:g_{\alpha}=g^{*}\\}$ is of size
$\mu^{+}$. Now $c(\alpha,f^{*}(i))$ depends only on the righthand coordinate
when we restrict ourselves to $H$ in the lefthand coordinate: if
$\alpha_{1},\alpha_{2}\in H$ then
$c(\alpha_{1},f^{*}(i))=g_{\alpha_{1}}(i)=g^{*}(i)=g_{\alpha_{2}}(i)=c(\alpha_{2},f^{*}(i))$.

The role of $\mathcal{M}$ and $f^{*}$ is to create the small component of the
monochromatic product. For the large component we choose a continuous
increasing sequence $\langle N_{\zeta}:\zeta\in\mu^{+}\rangle$ of submodels of
$\mathcal{H}(\chi)$ so that for every $\zeta\in\mu^{+}$ the following
requirements are met:

  1. $(\aleph)$

$\mu\subseteq N_{\zeta},|N_{\zeta}|=\mu$.

  2. $(\beth)$

$c,H,g^{*},\mathcal{M}\in N_{\zeta}$.

  3. $(\gimel)$

$N_{\zeta}\in N_{\zeta+1}$.

Notice that the set $E=\\{\zeta\in\mu^{+}:N_{\zeta}\cap\mu^{+}=\zeta\\}$ is a
club subset of $\mu^{+}$. It would be helpful to concentrate on members of
$E$. Since $E$ is a club we may assume, without loss of generality, that every
$\delta_{i}$ belongs to ${\rm acc}(E)$. For every $i<\kappa$ we choose an
ordinal $\alpha(i)\in\delta_{i}$ such that
$\alpha(i)>\bigcup\\{\nu_{j}:j<i\\}$. Also, pick $\alpha(*)\in H-\tau$.

The monochromatic product will be created now by induction. For every
$i<\kappa$ we build sets $A_{i},B_{i}$ which will approximate the large and
the small components of the product respectively. Likewise, we choose an
ordinal $j(i)\in\kappa$ and we try to keep the following requirements:

  1. $(a)$

$j(i)>i$; also, $i_{0}<i_{1}\Rightarrow\lambda_{j(i_{0})}<\mu_{j(i_{1})}$.

  2. $(b)$

If $\sigma_{0},\sigma_{1}\in\\{\delta_{\ell}:\ell\leq
i\\}\cup\\{\alpha(\ell):\ell\leq i\\}\cup\\{\alpha(*)\\}$ and
$\sigma_{0}<\sigma_{1}$ then $f_{\sigma_{0}}(j)<f_{\sigma_{1}}(j)$ for every
$j\in[j(i),\kappa)$.

  3. $(c)$

$A_{i}\subseteq H\cap\delta_{i},{\rm otp}(A_{i})=\delta_{i}$ and $A_{i}\in
M_{\nu_{i}j(i)}$.

  4. $(d)$

$B_{i}\subseteq\lambda_{j(i)}-\bigcup\\{\lambda_{j(\ell)}:\ell<i\\},{\rm
otp}(B_{i})=\lambda_{j(i)}$ and $B_{\ell}\in M_{\nu_{i}j(i)}$ for every
$\ell<i$.

  5. $(e)$

If $\alpha\in\bigcup\\{A_{\ell}:\ell\leq i\\}\cup\\{\alpha(*)\\}$ and
$\beta\in B_{\ell}$ for some $\ell\leq i$ then
$c(\alpha,\beta)=g^{*}(j(\ell))$.

Suppose that we can carry the induction. Define
$A=\bigcup_{i<\kappa}A_{i}\cup\\{\alpha(*)\\},B=\bigcup_{i<\kappa}B_{i}$.
Notice that $|B|=\bigcup\\{\lambda_{j(i)}:i<\kappa\\}=\mu$ and ${\rm
otp}(A)=\tau+1$. Choose any pair of ordinals $\alpha\in A,\beta\in B$. Let
$i<\kappa$ be the first ordinal so that $\beta<\lambda_{j(i)}$ and
$\alpha\in\bigcup\\{A_{\ell}:\ell\leq i\\}\cup\\{\alpha(*)\\}$. Requirement
$(d)$ implies that the $B_{\ell}$s are mutually disjoint. By the disjointness
of the $B_{\ell}$s there is a unique $\ell\leq i$ such that $\beta\in
B_{\ell}$, and then $c(\alpha,\beta)=g^{*}(j(\ell))$. This means that
$c(\alpha,\beta)$ depends only on the righthand coordinate (as $A\subseteq
H$), so by Lemma 2.1 we are done.

It remains to carry out the induction. Suppose that $i<\kappa$, and assume
that $A_{\ell},B_{\ell},j(\ell)$ have been defined for every $\ell<i$. We
shall choose $j(i)$ and build $A_{i}$ together, and then we shall describe the
construction of $B_{i}$. For every $\ell<i$ we pick an ordinal
$\eta_{\ell}\in\kappa$ such that $A_{\ell},B_{\ell},j(\ell)\in
M_{\alpha(i)\eta_{\ell}}$. Notice that for every $j\in[\eta_{\ell},\kappa)$ we
will have $A_{\ell},B_{\ell},j(\ell)\in M_{\alpha(i)j}$ by the properties of
the array $\mathcal{M}$. The choice of $\eta_{\ell}$ is possible since
$\nu_{\ell}<\alpha(i)$ and hence there exists $\eta_{\ell}\in\kappa$ such that
$M_{\nu_{\ell}j}\prec M_{\alpha(i)j}$ for every $j\in[\eta_{\ell},\kappa)$.
Now $A_{\ell},B_{\ell},j(\ell)\in M_{\nu_{\ell}j(\ell)}$ by the induction
hypothesis, so they belong to $M_{\nu_{\ell}j}$ for every $j\geq j(\ell)$. It
follows that for every sufficiently large $j<\kappa$ we have
$A_{\ell},B_{\ell},j(\ell)\in M_{\alpha(i)j}$. The above argument works
separately for every $\ell<i$, and we can choose now $j_{0}<\kappa$ such that
$A_{\ell},B_{\ell},j(\ell)\in M_{\alpha(i)j_{0}}$ simultaneously for every
$\ell<i$. This is possible since $i<\kappa$.

We choose an increasing sequence $\langle\zeta_{\varepsilon}:\varepsilon<{\rm
cf}(\delta_{i})\rangle$ of elements of $E$ such that $\zeta_{0}>\alpha(i)$ and
$\delta_{i}=\bigcup\\{\zeta_{\varepsilon}:\varepsilon<{\rm
cf}(\delta_{i})\\}$. Notice that $M_{\alpha(i)j_{0}}\prec
N_{\zeta_{\varepsilon}}$ for every $\varepsilon<{\rm cf}(\delta_{i})$: fixing
$\varepsilon<{\rm cf}(\delta_{i})$, we first note that $M_{\alpha(i)j_{0}}\in
N_{\zeta_{\varepsilon}}$, using that both ordinals $\alpha(i)$ and $j_{0}$
belong to $N_{\zeta_{\varepsilon}}$, as $\mathcal{M}\in
N_{\zeta_{\varepsilon}}$ and $j_{0}\in\kappa\subseteq
N_{\zeta_{\varepsilon}}$. Now, since $|M_{\alpha(i)j_{0}}|<\lambda_{j_{0}}$
(by the definition of a pcf array), we have in particular
$|M_{\alpha(i)j_{0}}|<\mu$; since $\mu\subseteq N_{\zeta_{\varepsilon}}$,
$M_{\alpha(i)j_{0}}\subseteq N_{\zeta_{\varepsilon}}$. Finally, as both
$M_{\alpha(i)j_{0}}$ and $N_{\zeta_{\varepsilon}}$ are elementary submodels of
${\mathcal{H}}(\chi)$ we may conclude that $M_{\alpha(i)j_{0}}\prec
N_{\zeta_{\varepsilon}}$.

Fix any $\varepsilon<{\rm cf}(\delta_{i})$. We define a first order formula
$\varphi(x)$ with parameters ($x$ is the free variable), as follows:

$x\in H\wedge x>\zeta_{\varepsilon}\wedge(\forall\ell<i)(\beta\in
B_{\ell}\rightarrow c(x,\beta)=g^{*}(j(\ell))).$

Remark that $\mathcal{H}(\chi)\models\varphi(\alpha(*))$ using the induction
hypothesis. Notice that $\varphi(x)$ is definable in
$N_{\zeta_{\varepsilon+1}}$, so by elementarity there is a set
$\Gamma_{\varepsilon}\subseteq
N_{\zeta_{\varepsilon+1}}-N_{\zeta_{\varepsilon}}$ such that ${\rm
otp}(\Gamma_{\varepsilon})=\zeta_{\varepsilon+1}$ and
$\gamma\in\Gamma_{\varepsilon}\Rightarrow\mathcal{H}(\chi)\models\varphi(\gamma)$.
Let $A_{i}=\bigcup\\{\Gamma_{\varepsilon}:\varepsilon<{\rm
cf}(\delta_{i})\\}$. Observe that ${\rm otp}(A_{i})=\delta_{i}$.

We can choose now $j(i)\in\kappa$ by setting the following requirements. First
we find $j_{1}\in\kappa$ such that $A_{i}\subseteq M_{\nu_{i}j_{1}}$ and then
we choose $j(i)\geq\max\\{j_{0},j_{1},i\\}$. Now we increase $j(i)$ if needed,
by considering a few more requirements. We make sure that $A_{i}\in
M_{\nu_{i}j_{1}}$. This can be done since $\delta_{i}<\nu_{i}$ and hence every
subset of $\delta_{i}$ is bounded in $\bigcup_{j\in\kappa}M_{\nu_{i}j}$. In
particular, $A_{i}\subseteq\delta_{i}$, so for some $j_{1}\in\kappa$ we see
that $A_{i}\subseteq M_{\nu_{i}j_{1}}\subseteq M_{\nu_{i}j(i)}$ and by
increasing the above chosen $j(i)$ we arrive at a sufficiently closed submodel
so that inclusion implies membership. If still needed we increase $j(i)$ once
more to get $M_{\nu_{i}j(i)}\prec M_{\alpha(*)j(i)}$ and
$f_{\delta_{i}}(j)<f^{*}(j)$ for every $j\geq j(i)$.

Finally, we define the set $B_{i}$. Let
$\Gamma=\\{\beta\in\lambda_{j(i)}:\forall\alpha\in\bigcup_{\ell\leq
i}A_{\ell}\cup\\{\alpha(*)\\},c(\alpha,\beta)=g^{*}(j(i))\\}$. Notice that
$\Gamma$ is definable in $M_{\alpha(*)j(i)}$, since all the parameters are in
this model. This is true, in particular, for $g^{*}$, since all the functions
from $\kappa$ to $\theta$ are in every $M_{\alpha i}$. The focal point here is
that $f^{*}(j(i))\in\Gamma$. Indeed, $f^{*}(j(i))<\lambda_{j(i)}$ and for
every $\alpha\in H$ we have $c(\alpha,f^{*}(j(i)))=g^{*}(j(i))$. This is true,
in particular, for every $\alpha\in\bigcup_{\ell\leq
i}A_{\ell}\cup\\{\alpha(*)\\}$ as all these ordinals are from $H$.

Since $f^{*}(j(i))>f_{\nu_{i}}(j(i))=\sup(M_{\nu_{i}j(i)}\cap\lambda_{j(i)})$
we infer that $f^{*}(j(i))\notin M_{\nu_{i}j(i)}$. By elementarity there is an
unbounded set of ordinals below $\lambda_{j(i)}$ in $M_{\nu_{i}j(i)}$ which
belong to $\Gamma$. Truncate the part of this set below
$\bigcup\\{\lambda_{j(\ell)}:\ell<i\\}$, and let $B_{i}$ be the remainder. One
can verify that $A_{i},B_{i}$ and $j(i)$ satisfy the requirements of the
inductive process, so we are done.

$\qed_{\ref{thmmt}}$

Let us add a clarification concerning singular cardinals of countable
cofinality. The representation theorem 1.1 is phrased in Shelah’s book [12]
with the assumption that $\mu>{\rm cf}(\mu)>\aleph_{0}$. In order to
incorporate singular cardinals of countable cofinality one can argue in two
different ways. Let us describe both strategies.

Firstly, one can try to improve the representation theorem in such a way that
singular cardinals with countable cofinality are included. Secondly, one can
argue that in the specific case of countable cofinality there is no need to
work with $J^{\rm bd}_{\omega}$, and every ideal $J\supseteq J^{\rm
bd}_{\omega}$ will give the desired result. Let us examine both alternatives.

For the first alternative, when the monograph of Shelah was published the
proof of Theorem 1.1 required uncountable cofinality. Later, Shelah improved
the methods of proof and obtained the same result at singular cardinals of
countable cofinality as well. Thus, in [13, Theorem 1.4] Shelah indicated
explicitly that a representation of $\mu^{++}$ with $J^{\rm bd}_{\omega}$
applies to $\mu>{\rm cf}(\mu)=\omega$ as well. We mention the fact that even
the classical representation of $\mu^{+}$ at singular cardinals with countable
cofinality is more involved, see [1, Exercise 2.25]. However, for $\mu^{++}$
(and even a bit more) one can still prove the same statement for all possible
cofinalities, including singular cardinals with countable cofinality.

The second alternative is not used in the current paper, but it is worth
mentioning this direction since it might be helpful in other statements of
this type. It is based on the fact that we need the specific ideal $J^{\rm
bd}_{\kappa}$ only at one point of the proof, where $\kappa={\rm cf}(\mu)$. We
deal with a sequence $\langle\lambda_{i}:i<\kappa\rangle$ so that
$\mu^{++}={\rm tcf}(\prod_{i<\kappa}\lambda_{i},J^{\rm bd}_{\kappa})$, but
then we may thin-out the sequence and we must be sure that the true cofinality
remains $\mu^{++}$. An arbitrary ideal does not necessarily preserve this
property, while $J^{\rm bd}_{\kappa}$ has this virtue. However, one needs only
the fact that $J^{\rm bd}_{\kappa}$ is $\kappa$-complete, and every
$\kappa$-complete ideal over $\kappa$ which extends $J^{\rm bd}_{\kappa}$ will
have the same effect.

In the specific case of $\kappa=\aleph_{0}$ we can modify the proof using any
ideal $J\subseteq J^{\rm bd}_{\omega}$, since every such ideal is
$\aleph_{0}$-complete (this is a special feature of $\aleph_{0}$, of course).
The changes in the proof are as follows. Rather than choosing $j(i)\in\kappa$
one has to choose a set $u_{i}\in{J}$ for every $i<\kappa$ and then let
$j(i)=\min(\kappa-u_{i})$. In all the arguments which apply to an end-segment
(i.e., for every $j\in[j(i),\kappa)$) one should focus on every $j\in\kappa-
u_{i}$. In the construction of the elementary submodels one should require
$u_{i}\in M_{\alpha(i)\eta_{\ell}}$ for each $i<\kappa$. Finally, instead of
choosing a sufficiently large $j$ by removing less than $\kappa$ many bounded
sets, one should pick an ordinal $j\in\kappa-\bigcup_{\ell\in{i}}u_{\ell}$.
This is possible by the $\kappa$-completeness of $J$ and the fact that
$u_{\ell}\in{J}$ for each $\ell\in{i}$.

In the original form of the paper we argued in this way, and the proof was
somewhat cumbersome. After a helpful discussion with Saharon, we realized that
there is no need to argue in that way, so we simplified the proof and
indicated that the required representation theorem works at singular cardinals
of countable cofinality as well. However, it might be useful in the future to
bear in mind that the argument can be based on $\kappa$-complete ideals in
general, and it is not necessary to employ $J^{\rm bd}_{\kappa}$.

In this paper we focused on the so-called _balanced_ polarized relation, in
which the required size of the monochromatic product is identical in both
colors. Let us add a few words with regard to the unbalanced relation. The
central assumption in our paper, as well as in [She98], is $2^{\mu}>\mu^{+}$.
However, if $\binom{\mu^{+}}{\mu}\rightarrow\binom{\tau}{\mu}_{\theta}$ for
every $\tau\in\mu^{+}$ and one collapses $2^{\mu}$ to $\mu^{+}$ then this
instance of an almost strong relation is preserved by virtue of the
completeness of the collapse.

Consider now the parallel unbalanced relation
$\binom{\mu^{+}}{\mu}\rightarrow\binom{\mu^{+}\quad\tau}{\mu\quad\mu}$. It was
shown in [GMS20] that if $\mu>{\rm cf}(\mu)=\omega$ and $2^{\mu}=\mu^{+}$ then
$\binom{\mu^{+}}{\mu}\nrightarrow\binom{\mu^{+}\quad\omega_{2}}{\mu\quad\mu}$,
and it was shown in [Gar21] that consistently
$\binom{\mu^{+}}{\mu}\nrightarrow\binom{\mu^{+}\quad\omega_{1}}{\mu\quad\mu}$.
For singular cardinals of countable cofinality this negative relation is
optimal, since for such cardinals
$\binom{\mu^{+}}{\mu}\rightarrow\binom{\mu^{+}\quad\tau}{\mu\quad\mu}$ for
every $\tau\in\omega_{1}$, as proved in [Jon08]. If ${\rm cf}(\mu)>\omega$
then the negative relation is even sharper, as
$\binom{\mu^{+}}{\mu}\nrightarrow\binom{\mu^{+}\quad\omega}{\mu\quad\mu}$, a
result of Erdős, Hajnal and Rado. Thus, assuming $2^{\mu}=\mu^{+}$ there is a
meaningful difference between the balanced and the unbalanced almost strong
polarized relations. This is reflected by the fact that at the balanced
relation one can obtain
$\binom{\mu^{+}}{\mu}\rightarrow\binom{\tau}{\mu}_{\theta}$ for every
$\tau\in\mu^{+},\theta\in{\rm cf}(\mu)$, and this is optimal. Indeed,
$2^{\mu}=\mu^{+}$ implies the failure of the strong relation, that is
$\binom{\mu^{+}}{\mu}\nrightarrow\binom{\mu^{+}}{\mu}$.

We conclude the paper with the following natural problem:

######  Question 2.3.

Let $\mu$ be a strong limit singular cardinal and suppose that
$2^{\mu}>\mu^{+}$. Is it consistent that
$\binom{\mu^{+}}{\mu}\nrightarrow\binom{\mu^{+}}{\mu}$?

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