On extremal numbers of the triangle plus the four-cycle

TL;DR

This paper introduces a new construction of dense graphs with girth at least five, improving the lower bound for the extremal number of graphs avoiding triangles and four-cycles, and resolving a longstanding open problem.

Contribution

It provides the first improvement on the lower bound of $ex(n,igracevert C_3,C_4 igracevert)$ since 1976, advancing extremal graph theory.

Findings

New dense graph constructions with girth ≥ 5

Improved lower bounds for extremal numbers of triangle and four-cycle free graphs

Negative answer to a problem posed by Chung-Graham

Abstract

For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function $ex(n,\{C_3,C_4\})$. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of $ex(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].