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

Abstract

In \cite{J}, Theorem 4.2, Jockusch proves that for any computable k-coloring of pairs of integers, there is an infinite ${\Pi }_2^0$ homogeneous set. The proof uses a countable collection of ${\Pi }_2^0$ 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 ${\Pi }_2^0$ sets. We provide a proof of this latter fact.