On idealized versions of $\pr_1(\mu^+,\mu^+,\mu^+,\cf(\mu))$

Abstract

We obtain an improvement of some coloring theorems from \cite{nsbpr}, \cite{819}, and \cite{APAL} for the case where the singular cardinal in question has countable cofinality. As a corollary, we obtain an "idealized" version of the combinatorial principle $\pr_1(\mu^+,\mu^+,\mu^+,\cf(\mu))$ that maximizes the indecomposability of the associated ideal.