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 König’s Lemma) stating that every infinite subtree of the full binary tree has an infinite path. We show that over $\mathsf {RCA}_0$, $\mathsf {TT}^2_k$ does not imply $\mathsf {WKL}_0$. This solves the open problem on the relative strength between the two major subsystems in second order arithmetic.

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中文翻译：

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树上成对的拉姆齐定理的强度：I. 弱 König 引理

摘要：令 $\mathsf {TT}^2_k$ 表示组合原理，即满二叉树中兼容节点对的每个 $k$-着色都有一个齐次解，即一个同构子树，其中所有兼容节点对有相同的颜色。令 $\mathsf {WKL}_0$ 是由基本系统 $\mathsf {RCA}_0$ 和原理（称为弱柯尼希引理）组成的二阶算术子系统，该原理表明全二叉树的每个无限子树都有一条无限的道路。我们证明在 $\mathsf {RCA}_0$ 上，$\mathsf {TT}^2_k$ 并不意味着 $\mathsf {WKL}_0$。这解决了二阶算法中两个主要子系统之间相对强度的开放问题。

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