Conservation 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.