Upper bounds on the extremal number of the 4-cycle

• DOI: 10.48550/arxiv.2107.11601
• Publication Date: 2021-01-01
• Authors: Jie Ma; Tianchi Yang
• Keywords: Extremal graph theory

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