# Polarized relations at singulars over successors

## TL;DR

This paper investigates the independence of certain polarized partition relations at singular cardinals over ZFC and ZF, showing that these relations cannot be proved or disproved within standard set theory.

## Contribution

It proves the independence of specific polarized partition relations at singular cardinals over ZFC and ZF, extending understanding of combinatorial set theory.

## Key findings

- Both relations are independent over ZFC.
- The relation is independent over ZF for some  of cofinality .
- Provides new insights into polarized relations at singulars.

## Abstract

Erdos, Hajnal and Rado asked whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_1}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\binom{\mu}{\aleph_2}\rightarrow\binom{\mu}{\aleph_2}_2$ is independent over ZF for some $\mu$ of cofinality $\omega_1$.