## Reverse mathematics and a Ramsey-type König’s lemma.(English)Zbl 1259.03022

In this paper a new principle, $$\mathsf{RKL}$$, is introduced and its strength, in the sense of reverse mathematics, is studied. $$\mathsf{RKL}$$ states that for an infinite binary tree $$T\subseteq 2^{<\mathbb N}$$ there is a set $$H$$ which is homogeneous in the sense that there are arbitrarily long $$\sigma\in T$$ with $$\sigma(x)=c$$ for all $$x\in H$$ with $$x< | \sigma|$$. (So in some sense the tree represents a coloring.) It is shown that $$\mathsf{RKL}$$ is a consequence of $$\mathsf{WKL}$$ and Ramsey’s theorem for pairs, $$\mathsf{RT}^2_2$$ (more precisely $$\mathsf{SRT}^2_2$$). Moreover, the author studies variants of this principle.

### MSC:

 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments
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### References:

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