Some upper bounds on ordinal-valued Ramsey numbers for colourings of pairs

TL;DR

This paper establishes upper bounds on ordinal-valued Ramsey numbers for pairs in two-colourings within the framework of $eta$-large sets, providing new insights into the logical strength of Ramsey's theorem for pairs.

Contribution

It introduces bounds on $eta$-large sets for Ramsey's theorem and offers a formalized proof that strengthens the understanding of its logical conservativity over $ extsf{RCA}_0$.

Findings

Any 2-colouring of pairs from an $oldsymbol{\omega^{300n} ext{-large}}$ set admits an $oldsymbol{\omega^n ext{-large}}$ homogeneous set.

A formalized version of the bound yields a more direct proof of the $oralloldsymbol{\Sigma^0_2}$-conservativity of Ramsey's theorem.

The results connect ordinal bounds with logical strength in reverse mathematics.

Abstract

We study Ramsey's theorem for pairs and two colours in the context of the theory of $\alpha$-large sets introduced by Ketonen and Solovay. We prove that any $2$-colouring of pairs from an $\omega^{300n}$-large set admits an $\omega^n$-large homogeneous set. We explain how a formalized version of this bound gives a more direct proof, and a strengthening, of the recent result of Patey and Yokoyama [Adv. Math. 330 (2018), 1034--1070] stating that Ramsey's theorem for pairs and two colours is $\forall\Sigma^0_2$-conservative over the axiomatic theory $\mathsf{RCA}_0$ (recursive comprehension).