Ramsey-like theorems and immunities

TL;DR

The paper investigates the computational complexity of Ramsey-like theorems, showing they are non-trivially complex by constructing specific colorings and analyzing their immunity properties.

Contribution

It proves that Ramsey-like theorems are not computably trivial and characterizes when they preserve certain immunity notions based on pattern shapes.

Findings

Constructed a computable 2-coloring where all pattern-avoiding infinite sets compute diagonally non-computable functions.

Characterized which Ramsey-like theorems preserve immunity notions based on pattern shape.

Contributes to the reverse mathematics understanding of Ramsey-like theorems.

Abstract

A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are computably trivial, by constructing a computable 2-coloring of $[\mathbb{N}]^2$ such that every infinite set avoiding any pattern computes a diagonally non-computable function relative to $\emptyset'$. We also consider multiple notions of weaknesses based of variants of immunity, and characterize the Ramsey-like theorems which preserve these notions or not, based on the shape of the avoided pattern. This is part of a larger study of the reverse mathematics of Ramsey-like theorems.