Ramsey numbers of degenerate graphs

TL;DR

This paper proves a bound on the Ramsey numbers of degenerate graphs with given chromatic number, confirming a conjecture from 1973 and establishing a relationship between degeneracy, chromatic number, and Ramsey numbers.

Contribution

It establishes a new exponential bound on the Ramsey number of d-degenerate graphs with specified chromatic number, solving a longstanding conjecture.

Findings

Ramsey number bound for d-degenerate graphs with chromatic number r

Confirmation of Burr and Erdős conjecture from 1973

Exponential relationship between graph parameters and Ramsey numbers

Abstract

A graph is $d$-degenerate if all its subgraphs have a vertex of degree at most $d$. We prove that there exists a constant $c$ such that for all natural numbers $d$ and $r$, every $d$-degenerate graph $H$ of chromatic number $r$ with $|V(H)| \ge 2^{d^22^{cr}}$ has Ramsey number at most $2^{d2^{cr}} |V(H)|$. This solves a conjecture of Burr and Erd\H{o}s from 1973.