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.


03B30 Foundations of classical theories (including reverse mathematics)
03F35 Second- and higher-order arithmetic and fragments
Full Text: DOI arXiv Euclid


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