An inside/outside Ramsey theorem and recursion theory

TL;DR

This paper analyzes the Rival-Sands inside/outside Ramsey theorem through reverse mathematics and Weihrauch degrees, revealing its computational strength and relation to other combinatorial principles, and establishing its precise place in the hierarchy.

Contribution

It provides the first detailed analysis of the Rival-Sands theorem's strength in reverse mathematics and Weihrauch degrees, showing its equivalence to arithmetical comprehension and the double jump of weak König's lemma.

Findings

Rival-Sands theorem is equivalent to arithmetical comprehension in reverse mathematics.

It is Weihrauch equivalent to the double jump of weak König's lemma.

The weak Rival-Sands theorem is weaker than Ramsey's theorem for pairs in Weihrauch degree.

Abstract

Inspired by Ramsey's theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph $G$ contains an infinite subset $H$ such that every vertex of $G$ is adjacent to precisely none, one, or infinitely many vertices of $H$. We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees. In reverse mathematics, we find that the Rival-Sands theorem is equivalent to arithmetical comprehension and hence is stronger than Ramsey's theorem for pairs. We also identify a weak form of the Rival-Sands theorem that is equivalent to Ramsey's theorem for pairs. We turn to the Weihrauch degrees to give a finer analysis of the Rival-Sands theorem's computational strength. We find that the Rival-Sands theorem is Weihrauch equivalent to the double jump of weak K\"{o}nig's lemma. We believe that the Rival-Sands theorem is the first natural theorem shown to exhibit exactly this strength. Furthermore, by combining our result with a result of Brattka and Rakotoniaina, we obtain that solving one instance of the Rival-Sands theorem exactly corresponds to simultaneously solving countably many instances of Ramsey's theorem for pairs. Finally, we show that the uniform computational strength of the weak Rival-Sands theorem is weaker than that of Ramsey's theorem for pairs by showing that a number of well-known consequences of Ramsey's theorem for pairs do not Weihrauch reduce to the weak Rival-Sands theorem. We also address an apparent gap in the literature concerning the relationship between Weihrauch degrees corresponding to the ascending/descending sequence principle and the infinite pigeonhole principle.