Stability and supersaturation of $4$-cycles

Jialin He, Jie Ma, Tianchi Yang

Published 2019-12-02Version 1

In this paper, we investigate extremal problems on the subject of the $4$-cycle $C_4$, which has played a heuristic important role in the development of extremal graph theory. As a milestone F\"uredi proved that the extremal number $ex(q^2+q+1, C_4)\leq \frac12 q(q+1)^2$ holds for $q\geq 14$. This matches with the fundamental construction of Erd\H{o}s-R{\'e}nyi-S\'os and Brown from finite geometry for prime powers $q$, thus providing one of the very rare exact results in the field. Our first result is a stability of F\"uredi's theorem. We prove that for all even $q$, every $(q^2+q+1)$-vertex $C_4$-free graph with at least $\frac12 q(q+1)^2-\frac12 q+o(q)$ edges must be a spanning subgraph of a unique polarity graph, which is constructed from a finite projective plane. Among others, this gives an immediate improvement on the upper bound of $ex(n,C_4)$ for infinite many integers $n$. A longstanding conjecture of Erd\H{o}s and Simonovits states that every $n$-vertex graph with $ex(n,C_4)+1$ edges contains at least $(1+o(1))\sqrt{n}$ copies of $C_4$. Building on the above stability, we obtain an exact result in this direction and thus confirm Erd\H{o}s-Simonovits conjecture for an infinite sequence of integers $n$. In fact our result implies a stronger assertion that generally speaking, whenever $n\gg \ell$ in these circumstances, we can characterize the graphs achieving the $\ell^{th}$ least number of $C_4$'s. This can be extended to more general settings, which provide enhancements on the supersaturation problem of $C_4$. We also discuss some related problems and formalize a conjecture on $ex(n,C_4)$, whose affirmation would disprove Erd\H{o}s-Simonovits conjecture for general $n$.