Aristotelian poetry

TL;DR

The paper proves a set-theoretic consistency result related to polarized partition relations, reducing the required large cardinal assumption to an omega_1-Erdos cardinal.

Contribution

It establishes the consistency of a polarized partition relation with a lower large cardinal assumption, refining previous results.

Findings

Consistency of $inom{oldsymbol{ ext{omega}_2}}{oldsymbol{ ext{omega}_1}} ightarrow inom{n}{oldsymbol{ ext{omega}_1}}_oldsymbol{ extomega}$ for all n in omega.

Negative relation $inom{oldsymbol{ extomega}_2}{oldsymbol{ extomega}_1} rightarrow inom{oldsymbol{ extomega}}{oldsymbol{ extomega}_1}_oldsymbol{ extomega}$ holds.

Reduced the large cardinal strength needed to prove the consistency.

Abstract

Jing Zhang proved the consistency of $\binom{\omega_2}{\omega_1}\rightarrow\binom{n}{\omega_1}_\omega$ for every $n\in\omega$ with the negative relation $\binom{\omega_2}{\omega_1}\nrightarrow\binom{\omega}{\omega_1}_\omega$. We reduce the consistency strength of this statement to an $\omega_1$-Erdos cardinal.