Ramsey-like theorems for separable permutations

TL;DR

This paper explores Ramsey-like theorems related to separable permutations, revealing their unique role in the existence of infinite homogeneous sets and developing new methods for relativized diagonal non-computation.

Contribution

It establishes the equivalence between avoiding separable permutations and the existence of infinite homogeneous sets, introducing a novel argument for relativized diagonal non-computation.

Findings

Avoidance of separable permutations implies infinite homogeneous sets.

This property does not hold for non-separable patterns.

Develops a new approach for relativized diagonal non-computation.

Abstract

We conduct a computability-theoretic study of Ramsey-like theorems of the form "Every coloring of the edges of an infinite clique admits an infinite sub-clique avoiding some pattern", with a particular focus on transitive patterns. As it turns out, the patterns corresponding to separable permutations play an important role in the computational features of the statement. We prove that the avoidance of any separable permutation is equivalent to the existence of an infinite homogeneous set in standard models, while this property fails for any other pattern. For this, we develop a novel argument for relativized diagonal non-computation.