Strong polarized relations for the continuum

TL;DR

This paper demonstrates that strong polarized relations for the continuum are valid for countable and supercompact cardinals using Mathias forcing iterations.

Contribution

It establishes the validity of strong polarized relations for the continuum at specific large cardinal levels with a novel forcing approach.

Findings

Strong polarized relations hold for  and supercompact cardinals.

Mathias forcing iteration is effective in proving these relations.

Results extend the understanding of continuum relations at large cardinals.

Abstract

We prove that the strong polarized relation for the continuum holds for $\aleph_0$ and for every supercompact cardinal. We use iteration of Mathias forcing.