Ramsey's Theorem for Pairs and k Colors as a Sub-Classical Principle of Arithmetic

Berardi, Stefano; Steila, Silvia (2017). Ramsey's Theorem for Pairs and k Colors as a Sub-Classical Principle of Arithmetic. The journal of symbolic logic, 82(02), pp. 737-753. Cambridge University Press 10.1017/jsl.2016.41

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The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of k-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number k ≥ 2, Ramsey's Theorem for pairs and recursive assignments of k colors is equivalent to the Limited Lesser Principle of Omniscience for Σ 0 3 formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinite k-ary tree there is some i < k and some branch with infinitely many children of index i .

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