Stable Ramsey's theorem and measure
Damir D. Dzhafarov

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 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 or in the context of reverse mathematics.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Mathematical and Theoretical Analysis
