Independent arithmetic progressions

TL;DR

This paper proves that sparse graphs contain large independent arithmetic progressions and explores implications for Ramsey theory, establishing a new link between graph sparsity and structured independent sets.

Contribution

It introduces a bound on graph edges ensuring large independent arithmetic progressions and applies this to solve problems in Ramsey theory.

Findings

Sparse graphs have large independent arithmetic progressions.

Established a bound on edges for such progressions to exist.

Applied results to various Ramsey theory questions.

Abstract

We show that there is a positive constant $c$ such that any graph on vertex set $[n]$ with at most $c n^2/k^2 \log k$ edges contains an independent set of order $k$ whose vertices form an arithmetic progression. We also present applications of this result to several questions in Ramsey theory.