Disproof of a conjecture of Erd\H{o}s and Simonovits on the Tur\'an number of graphs with minimum degree 3

TL;DR

This paper disproves a longstanding conjecture by Erdős and Simonovits regarding the Turán number of bipartite graphs with minimum degree 3, showing that the conjecture does not hold for all such graphs.

Contribution

The paper constructs specific 3-regular bipartite graphs with Turán numbers exceeding the conjectured bound, providing a counterexample to the conjecture.

Findings

Disproved the conjecture for 3-regular bipartite graphs.

Constructed graphs with Turán number O(n^{4/3+ε}) for any ε>0.

Showed the conjecture does not hold universally for bipartite graphs with minimum degree 3.

Abstract

In 1981, Erd\H{o}s 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, Erd\H{o}s offered 250 dollars for a proof and 500 dollars for a counterexample. In this paper, we disprove the conjecture by finding, for any $\varepsilon>0$, a $3$-regular bipartite graph $H$ with $\mathrm{ex}(n,H)=O(n^{4/3+\varepsilon})$.