On "finitary" Ramsey's theorem

TL;DR

This paper explores a 'finitary' version of Ramsey's theorem, connecting it to various foundational results in logic and combinatorics, including well-foundedness and consistency of formal theories.

Contribution

It introduces a 'finitary' infinite Ramsey's theorem and demonstrates its implications for well-foundedness, consistency, and classical Ramsey results.

Findings

Establishes a connection between 'finitary' Ramsey's theorem and well-foundedness of certain ordinals.

Shows that the 'finitary' version implies 1-consistency of PA.

Links the 'finitary' theorem to classical finite and infinite Ramsey theorems.

Abstract

We examine a version of Ramsey's theorem based on Tao, Gaspar and Kohlenbach's "finitary" infinite pigeonhole principle.We will show that the "finitary" infinite Ramsey's theorem naturally gives rise to statements at the level of the infinite Ramsey's theorem, Friedman's infinite adjacent Ramsey theorem (well-foundedness of certain ordinals up to $\varepsilon_0$), $1$-consistency of theories up to PA and the finite Ramsey's theorem.