Partition Calculus and Cardinal Invariants

TL;DR

This paper demonstrates the consistency of strong polarized relations across a range of cardinals and their implications for continuum cardinal invariants, extending results to large cardinals and singular limits.

Contribution

It establishes the consistency of strong polarized relations for all cardinals in a given interval and explores their implications for continuum invariants, including large and singular cardinals.

Findings

Consistency of strong polarized relations for all cardinals in [,]

Application to continuum cardinal invariants

Extension to supercompact and singular cardinals

Abstract

We prove that the strong polarized relation of $\theta$ above $\omega$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal invariant on the continuum. We show that similar results hold for a supercompact cardinal, and for a strong limit singular under some assumptions.