Coloring the rationals in reverse mathematics

TL;DR

This paper investigates a strengthened form of Ramsey's theorem for pairs, focusing on colorings of rationals, and explores its logical strength within reverse mathematics.

Contribution

It introduces and analyzes the Erdos-Rado theorem for rationals, positioning it within the hierarchy of reverse mathematical principles.

Findings

The Erdos-Rado theorem for rationals is a natural candidate between ACA and RT^2.

It belongs to a family of Ramsey-type statements with challenging logical strength.

The theorem's strength relates to the complexity of homogeneous sets in rational orderings.

Abstract

Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erdos and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type eta, where eta is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey's theorem for pairs. This Erdos-Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.