An almost strong relation

TL;DR

The paper tackles almost strong polarized partition relations at strong limit singular cardinals. It develops a pcf-array framework of elementary submodels to realize a representation of $\mu^{++}$ as a true cofinality of a product $\prod_{i<\kappa}\lambda_i$ modulo $J^{\rm bd}_\kappa$, enabling a uniform bound on characteristic functions and enabling control over monochromatic sets. Under $\mu>\mathrm{cf}(\mu)=\kappa$ with $\mu$ strong limit and $2^\mu>\mu^+$, it proves $\binom{\mu^+}{\mu} \rightarrow \binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ for every $\tau<\mu^+$, extending previous results of Erd\H{o}s–Hajnal–Rado and Shelah to uncountable cofinality and general $\tau$. Moreover, when $2^\mu=\mu^+$, the authors show that the almost strong relation is preserved after collapsing $2^\mu$ to $\mu^+$, yielding an optimal positive relation and highlighting a meaningful distinction between balanced and unbalanced polarized relations. The approach blends pcf theory with a new feature that stretches monochromatic sets, potentially guiding future applications of pcf arrays beyond this result.

Abstract

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

Figures (1)

• Figure 1: A pcf-array

Theorems & Definitions (6)

• Theorem 1.1
• Definition 1.2: Pcf array
• Definition 1.3: The characteristic sequence
• Claim 1.4
• Lemma 2.1
• Theorem 2.2