An almost strong relation

TL;DR

This paper proves a new combinatorial relation involving strong limit singular cardinals, establishing conditions under which certain partition relations hold, and demonstrates the optimality of these relations under specific cardinal arithmetic assumptions.

Contribution

It introduces a new partition relation for strong limit singular cardinals and shows its optimality under the assumption that 2^μ equals μ^+.

Findings

Proves a partition relation for strong limit singular cardinals when 2^μ > μ^+.

Establishes the optimality of the relation under 2^μ = μ^+ after collapsing.

Demonstrates the preservation of the positive relation under certain forcing conditions.

Abstract

Let $\mu$ be a strong limit singular cardinal. We prove that if $2^{\mu} > \mu^+$ then $\binom{\mu^+}{\mu}\to \binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ 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.