Ramsey-like theorems for the Schreier barrier

TL;DR

This paper generalizes classical combinatorial theorems to the Schreier barrier, analyzing their logical strength and computational complexity, revealing surprising coding capabilities related to the arithmetical hierarchy.

Contribution

It formulates and proves generalized Ramsey-type theorems for the Schreier barrier and analyzes their strength in computability and reverse mathematics.

Findings

Exactly ω-large Thin Set and Free Set theorems can code ^{()}.

Exactly -large Rainbow Ramsey theorem does not code the halting set.

Theorems' strength varies significantly in computational and logical frameworks.

Abstract

The family of finite subsets $s$ of the natural numbers such that $|s|=1+\min s$ is known as the Schreier barrier in combinatorics and Banach Space theory, and as the family of exactly $\omega$-large sets in Logic. We formulate and prove the generalizations of Friedman's Free Set and Thin Set theorems and of Rainbow Ramsey's theorem to colorings of the Schreier barrier. We analyze the strength of these theorems from the point of view of Computability Theory and Reverse Mathematics. Surprisingly, the exactly $\omega$-large counterparts of the Thin Set and Free Set theorems can code $\emptyset^{(\omega)}$, while the exactly $\omega$-large Rainbow Ramsey theorem does not code the halting set.