Ramsey's theorem and cone avoidance

It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low 2 homogeneous set H . We answer a question of the same authors by showing that H may be chosen to satisfy in addition C ≰ r H , where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring of pairs admits a pair of low 2 infinite homogeneous sets whose degrees form a minimal pair.