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

Abstract

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 $k \geq 2$, Ramsey's Theorem for pairs and recursive assignments of $k$ colors is equivalent to the Limited Lesser Principle of Omniscience for $\Sigma^0_3$ 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 $i$.