Free sets, thin sets and rainbows for barriers

TL;DR

This paper generalizes classical combinatorial theorems to barriers, analyzing their logical and computational complexity, and establishing bounds and reductions within reverse mathematics.

Contribution

It introduces generalized versions of free set, thin set, and rainbow Ramsey theorems for barriers, with complexity and proof-theoretic analyses.

Findings

Established upper and lower bounds on solution complexity for computable instances.

Proved uniform computable reductions between theorems.

Derived proof-theoretic results on the logical strength of these theorems.

Abstract

We formulate and prove the generalizations of Friedman's free set and thin set theorems and of the rainbow Ramsey theorem to colorings of barriers. We analyze the strength of these theorems from the point of view of computability theory proving some upper and lower bounds on the complexity of solutions for computable instances and some uniform computable reductions. We obtain as corollaries some proof-theoretical results on the logical strength of the theorems, in the spirit of reverse mathematics.