A packed Ramsey's theorem and computability theory

TL;DR

This paper introduces a variant of Ramsey's theorem called the packed Ramsey's theorem, analyzing its computational and logical strength, and establishing its equivalence to standard Ramsey's theorem for certain exponents.

Contribution

It characterizes the computational and reverse mathematical strength of the packed Ramsey's theorem, linking it to classical Ramsey's theorem for specific exponents.

Findings

Equivalent to Ramsey's theorem for exponents not equal to 2 in reverse mathematics

Implicates Ramsey's theorem when n=2 but does not imply ACA_0

Provides bounds for solutions to computable instances

Abstract

Ramsey's theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out. Paul Erdos and Fred Galvin proved that for each coloring f, there is an infinite set that is "packed together" which is given "a small number" of colors by f. We analyze the strength of this theorem from the perspective of computability theory and reverse mathematics. We show that this theorem is close in computational strength to standard Ramsey's theorem by giving arithmetical upper and lower bounds for solutions to computable instances. In reverse mathematics, we show that that this packed Ramsey's theorem is equivalent to Ramsey's theorem for exponents not equal to 2. When n=2, we show that it implies Ramsey's theorem, and that it does not imply ACA_0.