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

TL;DR

This paper investigates the logical strength of Ramsey's Theorem for pairs with k colors within intuitionistic arithmetic, establishing equivalences with certain limited omniscience principles and properties of infinite recursive trees.

Contribution

It demonstrates the equivalence between Ramsey's Theorem for pairs with recursive k-colorings and specific principles of limited omniscience in intuitionistic arithmetic.

Findings

Equivalence with the Limited Lesser Principle of Omniscience for -formulas

Connection to branches in infinite recursive k-ary trees

Results hold for all natural numbers k  2

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 $i$.