The Strength of Ramsey's Theorem For Pairs over trees: I. Weak K\"onig's Lemma

Abstract

Let $\mathsf{TT}^2_k$ denote the combinatorial principle stating that every $k$-coloring of pairs of compatible nodes in the full binary tree has a homogeneous solution, i.e. an isomorphic subtree in which all pairs of compatible nodes have the same color. Let $\mathsf{WKL}_0$ be the subsystem of second order arithmetic consisting of the base system $\mathsf{RCA}_0$ together with the principle (called Weak K\"onig's Lemma) stating that every infinite subtree of the full binary tree has an infinite path. We show that over $\mathsf{RCA}_0$, $\mathsf{TT}^2_k$ doe not imply $\mathsf{WKL}_0$. This solves the open problem on the relative strength between the two major subsystems of second order arithmetic.