Determinacy and J\'onsson cardinals in $L(\mathbb{R})$

TL;DR

Under ZF+AD+V=L(R), the paper proves uncountable cardinals below Theta are Jnsson, with those of cofinality omega being Rowbottom, and explores related partition properties.

Contribution

The paper establishes that in this set-theoretic framework, certain large cardinals have specific partition properties, extending understanding of their structure.

Findings

Uncountable cardinals below Theta are Jnsson.

If cof(kappa)=omega, then kappa is Rowbottom.

Additional partition properties are demonstrated.

Abstract

Assume $\mathsf{ZF}+\mathsf{AD}+V=L(\mathbb{R})$ and let $\kappa<\Theta$ be an uncountable cardinal. We show that $\kappa$ is J\'onsson, and that if $\mathrm{cof}(\kappa)=\omega$ then $\kappa$ is Rowbottom. We also establish some other partition properties.