Disproof of a Conjecture of Erdős and Simonovits on the Turán Number of Graphs with Minimum Degree 3

Abstract

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