Using the Incompressibility Method to obtain Local Lemma results for Ramsey-type Problems

TL;DR

This paper introduces a novel approach connecting the incompressibility method with the Lovasz local lemma to derive bounds in Ramsey theory, demonstrated on van der Waerden numbers and applicable to various Ramsey problems.

Contribution

It establishes a new link between incompressibility and the Lovasz local lemma, providing a method to obtain bounds in Ramsey-type problems.

Findings

Derived bounds for van der Waerden numbers using the method

Applicable to lower bounds of Ramsey numbers and transitive subtournaments

Demonstrated the method's versatility across multiple Ramsey phenomena

Abstract

We reveal a connection between the incompressibility method and the Lovasz local lemma in the context of Ramsey theory. We obtain bounds by repeatedly encoding objects of interest and thereby compressing strings. The method is demonstrated on the example of van der Waerden numbers. It applies to lower bounds of Ramsey numbers, large transitive subtournaments and other Ramsey phenomena as well.