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

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

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