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If we look at reverse mathematics statements as mass problems, considering the class of solutions of an instance, it is known that Weak König's lemma has a maximal instance in the sense that there is an infinite computable binary tree $T$ such that for every path $P$ in the tree and every infinite computable binary tree $U$, $P$ computes a path in $U$.

I heard that it was known not to be the case for Ramsey theorem for pairs, ie for every computable coloring function $f$ over pairs, there is an infinite homogeneous set $H$ and a computable coloring function $g$ over pairs such that $H$ computes no infinite homogeneous set for $g$. But I couldn't find any reference. Does someone know who proved it ?

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up vote 5 down vote accepted

This is due to Joe Mileti and can be found in his thesis, Corollary 5.4.7.

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