Upper bounds on the extremal number of the 4-cycle

TL;DR

This paper establishes new upper bounds on the maximum edges in n-vertex graphs avoiding 4-cycles, disproving a longstanding conjecture and providing asymptotically optimal bounds for many n.

Contribution

It introduces improved upper bounds on the extremal number for 4-cycle-free graphs, challenging Erdős's 1970s conjecture and advancing understanding of extremal graph theory.

Findings

New upper bounds on f(n) for 4-cycle-free graphs

Disproof of Erdős's conjecture from the 1970s

Asymptotically optimal bounds for a broad range of n

Abstract

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a disproof of a conjecture of Erd\H{o}s from 1970s which asserts that $f(n)=\frac12 n^{3/2}+\frac14 n+o(n)$.