Recursion theoretic aspects of the dual ramsey theorem

Abstract

Ramsey's Theorem deals with colorings of the k-element subsets of a fixed infinite set. The Dual Ramsey Theorem deals with colorings of the k-element partitions of a fixed infinite set. We review the statement of the Dual Ramsey Theorem and of its generalization to A-partitions. Here A is a finite alphabet. We discuss the problem of optimal recursion-theoretic bounds for the Dual Ramsey Theorem. We point out applications of the Dual Ramsey Theorem to initial segment constructions in degree theory. We state a result of Brackin concerning partitions which have no coarsenings of higher Turing degree. In an appendix we summarize the known results (and present two new results) concerning nearly recursive (i.e. bi-immune free) Turing degrees.