Many partition relations below density

TL;DR

This paper demonstrates how to force large continuum values while establishing strong polarized partition relations for many pairs in the interval, with applications to problems in general topology.

Contribution

It introduces a method to simultaneously increase the continuum and establish strong polarized partition relations in a broad interval, advancing understanding in set theory and topology.

Findings

Established strong polarized partition relations for many pairs in (rac{,2^})

Showed how to force large continuum values while maintaining these relations

Applied results to problems in general topology

Abstract

We force $2^\lambda$ to be large and for many pairs in the interval $(\lambda,2^\lambda)$ a stronger version of the polarized partition relations hold. We apply this toproblem in general topology