Conservation Strength of The Infinite Pigeonhole Principle for Trees

TL;DR

This paper proves that the combinatorial principle $ ext{TT}^1$, related to infinite binary trees, is conservatively equivalent to Ramsey's theorem for pairs over a weak base system, in terms of $ ext{Pi}^0_3$ sentences.

Contribution

It establishes the $ ext{Pi}^0_3$-conservativity of $ ext{TT}^1$ combined with other principles over $ ext{RCA}_0$, showing their logical equivalence in a specific fragment.

Findings

$ ext{TT}^1 + ext{RT}^2_2 + ext{WKL}_0$ is $ ext{Pi}^0_3$-conservative over $ ext{RCA}_0$

$ ext{TT}^1$ and Ramsey's theorem for pairs prove the same $ ext{Pi}^0_3$-sentences over $ ext{RCA}_0$

The result clarifies the logical strength of the infinite pigeonhole principle for trees.

Abstract

Let $\mathsf{TT}^1$ be the combinatorial principle stating that every finite coloring of the infinite full binary tree has a homogeneous isomorphic subtree. Let $\mathsf{RT}^2_2$ and $\mathsf{WKL}_0$ denote respectively the principles of Ramsey's theorem for pairs and weak K\"onig's lemma. It is proved that $\mathsf{TT}^1+\mathsf{RT}^2_2+\mathsf{WKL}_0$ is $\Pi^0_3$-conservative over the base system $\mathsf{RCA}_0$. Thus over $\mathsf{RCA}_0$, $\mathsf{TT}^1$ and Ramsey's theorem for pairs prove the same $\Pi^0_3$-sentences.