Disproof of a conjecture of Erdős and Simonovits on the Turán number of graphs with minimum degree 3

Abstract

In 1981, Erdős and Simonovits conjectured that for any bipartite graph $H$ we have $\mathrm{ex}(n,H)=O(n^{3/2})$ if and only if $H$ is $2$-degenerate. Later, Erdős offered 250 dollars for a proof and 500 dollars for a counterexample. In this paper, we disprove the conjecture by finding, for any $\epsilon >0$, a $3$-regular bipartite graph $H$ with $\mathrm{ex}(n,H)=O(n^{4/3+\epsilon })$.

arXiv comment12 pages; v2 contains additional references