Ramsey numbers of degenerate graphs

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 Erdős from 1973.