# Ramsey-like theorems and moduli of computation

## TL;DR

This paper introduces a class of Ramsey-like theorems that generalize many reverse mathematics results, showing that such theorems compute Turing degrees primarily through the sparsity of solutions.

## Contribution

It defines a maximal Ramsey-like statement satisfying cone avoidance and uses it to unify and reprove many existing results in reverse mathematics.

## Key findings

- Identifies a maximal cone avoidance property in Ramsey-like theorems.
- Shows that solutions depend only on the sparsity of elements.
- Provides a criterion to reprove known cone avoidance results.

## Abstract

Ramsey's theorem asserts that every $k$-coloring of $[\omega]^n$ admits an infinite monochromatic set. Whenever $n \geq 3$, there exists a computable $k$-coloring of $[\omega]^n$ whose solutions compute the halting set. On the other hand, for every computable $k$-coloring of $[\omega]^2$ and every non-computable set $C$, there is an infinite monochromatic set $H$ such that $C \not \leq_T H$. The latter property is known as cone avoidance. In this article, we design a natural class of Ramsey-like theorems encompassing many statements studied in reverse mathematics. We prove that this class admits a maximal statement satisfying cone avoidance and use it as a criterion to re-obtain many existing proofs of cone avoidance. This maximal statement asserts the existence, for every $k$-coloring of $[\omega]^n$, of an infinite subdomain $H \subseteq \omega$ over which the coloring depends only on the sparsity of its elements. This confirms the intuition that Ramsey-like theorems compute Turing degrees only through the sparsity of its solutions.