The canonical Ramsey theorem and computability theory

Abstract

Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and König’s Lemma.