Complete Bipartite Partition Relations in Cohen Extensions

TL;DR

This paper studies how adding Cohen reals affects partition relations on _2, showing certain relations hold after forcing but not others, revealing the optimality of these results.

Contribution

It demonstrates the effect of Cohen forcing on bipartite partition relations on _2, establishing both positive and negative results in this context.

Findings

_2  (_2,  : )^2 holds after Cohen forcing.

Cohen forcing does not preserve _2  (_2,  : _1)^2.

Results are optimal within the studied forcing extension.

Abstract

We investigate the effect of adding $\omega_2$ Cohen reals on graphs on $\omega_2$, in particular we show that $\omega_2 \to (\omega_2, \omega : \omega)^2$ holds after forcing with $\mathsf{Add}(\omega, \omega_2)$ in a model of $\mathsf{CH}$. We also prove that this results is in a certain sense optimal as $\mathsf{Add}(\omega, \omega_2)$ forces that $\omega_2 \not\to (\omega_2, \omega : \omega_1)^2$.