Ramsey Theorem for Pairs As a Classical Principle in Intuitionistic Arithmetic

We produce a first order proof of a famous combinatorial result, Ramsey Theorem for pairs and in two colors. Our goal is to find the minimal classical principle that implies the “miniature” version of Ramsey we may express in Heyting Arithmetic HA. We are going to prove that Ramsey Theorem for pairs with recursive assignments of two colors is equivalent in HA to the sub-classical principle Σ3-LLPO. One possible application of our result could be to use classical realization to give constructive proofs of some combinatorial corollaries of Ramsey; another, a formalization of Ramsey in HA, using a proof assistant. In order to compare Ramsey Theorem with first order classical principles, we express it as a schema in the first order language of arithmetic, instead of using quantification over sets and functions as it is more usual: all sets we deal with are explicitly defined as arithmetical predicates. In particular, we formally define the homogeneous set which is the witness of Ramsey theorem as a ∆3-arithmetical predicate. 1998 ACM Subject Classification F.4.1 Mathematical Logic