Suppose we have an "oracle" that gives a homogeneous set for a 2-coloring $\hat c : [\omega]^2 \rightarrow 2$ of pairs of integers. Using this oracle, can we "compute" a homogeneous set for a 3-coloring $c : [\omega]^2\rightarrow 3$ for pairs of integers? The answer is an obvious yes - you can use the oracle twice to do that. But what if we are allowed to use the oracle only once?
More technically, for every 3-coloring $c : [\omega]^2 \rightarrow 3$, does there exist a 2-coloring $\hat c : [\omega]^2\rightarrow 2$, computable from $c$, such that for any homogeneous set $\hat H$ for $\hat c$ there exists a homogeneous set $H$ for $c$ computable from $\hat H$ and $c$?
Background : It is a "fact", supposedly easy to prove, stated in a paper by Hirschfeldt and Jockusch, that such $\hat c$ exists. Addendum It's at the bottom of page 34.