Cross-constraint basis theorems and products of partitions

We survey and extend a new technique from Lu Liu to prove separation theorems between products of Ramsey-type theorems over computable reducibility. We use this technique to show that Ramsey’s theorem for n -tuples and three colors is not computably reducible to finite products of Ramsey’s theorem for n -tuples and two colors, and prove multiple basis and anti-basis theorems for the so-called cross-constraint problem.