Open and solved problems concerning polarized partition relations

TL;DR

This paper discusses open problems in polarized partition relations and solves some by demonstrating the ability to force strong polarized relations at singular cardinals.

Contribution

It introduces new solutions to open problems in polarized partition relations, particularly at singular cardinals, expanding understanding in this area.

Findings

Solved open problems by forcing strong polarized relations at singular cardinals.

Demonstrated the consistency of certain polarized partition relations.

Extended the theoretical framework of polarized partition relations.

Abstract

We list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $\mu$ one can force the strong polarized relation with respect to the pair $\mu^+,\mu$.