Computable Ramsey's theorem for pairs needs infinitely many \(\Pi ^0_2\) sets (Q512143)

\textit{C. G. Jockusch jun.} [J. Symb. Log. 37, 268--280 (1972; Zbl 0262.02042)] initiated the study of the effective content of Ramsey's theorem and proved that for any computable \(k\)-coloring of pairs of integers, there is an infinite \(\Pi_2^0\) homogeneous set. The proof used a countable collection of \(\Pi_2^0\) sets as potential infinite homogeneous sets. He stated 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. The authors prove the claim, showing that there is no computable way to take an index for an arbitrary computable coloring and produce a finite number of indices of \(\Pi_2^0\) sets with the property that one of those sets will be homogeneous for that coloring. For proving this result, the authors introduce \(n\)-trains as objects with useful combinatorial properties which can be used as approximations to infinite \(\Pi_2^0\) sets.