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

TL;DR

This paper improves coloring theorems for singular cardinals with countable cofinality and introduces an idealized version of a combinatorial principle that enhances ideal indecomposability.

Contribution

It provides an improved version of existing coloring theorems and formulates an idealized combinatorial principle maximizing ideal indecomposability.

Findings

Enhanced coloring theorems for singular cardinals with countable cofinality

An idealized version of the principle (,,, ext{cf}())

Maximized indecomposability of the associated ideal

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.