An almost strong relation

Abstract

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