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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 ahomogeneous solution, i.e. an isomorphic subtree in which all pairs ofcompatible nodes have the same color. Let $\mathsf{WKL}_0$ be the subsystem ofsecond order arithmetic consisting of the base system $\mathsf{RCA}_0$ togetherwith the principle (called Weak K\"onig's Lemma) stating that every infinitesubtree 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 openproblem on the relative strength between the two major subsystems of secondorder arithmetic.

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