Computable Ramsey's Theorem for Pairs Needs Infinitely Many Pi-0-2 Sets

TL;DR

This paper proves that for computable colorings of pairs, infinitely many Pi-0-2 sets are necessary to find an infinite homogeneous set, showing a fundamental limitation in computable combinatorics.

Contribution

It establishes that a finite collection of Pi-0-2 sets cannot suffice, confirming a conjecture about the complexity of homogeneous sets in computable Ramsey theory.

Findings

Finite Pi-0-2 sets are insufficient for homogeneous sets

Infinite Pi-0-2 sets are necessary in the computable setting

Confirms a conjecture by Jockusch about the limitations of finite collections

Abstract

In \cite{J}, Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite $\Pi^0_2$ homogeneous set. The proof uses a countable collection of $\Pi^0_2$ sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch states without proof that it can be shown that there is no computable way to prove this result with a finite number of $\Pi^0_2$ sets. We provide a proof of this latter fact.