Stable Ramsey's theorem and measure

TL;DR

This paper introduces a measure-theoretic restriction of the stable Ramsey's theorem for pairs, analyzing its computational strength and its relation to other principles in reverse mathematics.

Contribution

It defines a weaker form of the stable Ramsey's theorem based on measure-theoretic non-null subclasses and investigates its implications and limitations.

Findings

Sets computing infinite homogeneous sets for non-null stable colorings align with those for all stable colorings below '

The weaker principle does not imply coh or wkl_0 in reverse mathematics

The computational power for non-null and all stable colorings differs above f'

Abstract

The stable Ramsey's theorem for pairs has been the subject of numerous investigations in mathematical logic. We introduce a weaker form of it by restricting from the class of all stable colorings to subclasses of it that are non-null in a certain effective measure-theoretic sense. We show that the sets that can compute infinite homogeneous sets for non-null many computable stable colorings and the sets that can compute infinite homogeneous sets for all computable stable colorings agree below $\emp'$ but not in general. We also answer the analogs of two well known questions about the stable Ramsey's theorem by showing that our weaker principle does not imply $\mathsf{COH}$ or $\mathsf{WKL}_0$ in the context of reverse mathematics.