Some exact results on $4$-cycles: stability and supersaturation
Jialin He, Jie Ma, Tianchi Yang
TL;DR
The paper advances exact extremal results for the 4-cycle by establishing a strong stability result: for large even $q$, any $C_4$-free graph on $q^2+q+1$ vertices with nearly maximal edges embeds in a unique polarity graph, sharpening Füredi’s upper bound and informing ex$(n,C_4)$ for infinitely many $n$. It develops a novel constructive approach linking $C_4$-free graphs to polarity graphs via large 1-intersecting hypergraphs and projective-plane embeddings, enabling an exact supersaturation description: for $n=q^2+q+1$ and added edges $t$, either many $C_4$ copies occur, or the graph is a polarity-graph plus $t$ edges with a precise copy-count in the range $[sq-s^2, sq+s^2]$. The results yield exact or near-optimal bounds for ex$(n,C_4)$ and h$(n,t)$ on infinite families of $n$, and they provide a refined understanding of how extremal and supersaturated configurations are distributed around polarity-graph structures. These findings bridge extremal graph theory with finite geometry, offering exact classifications of extremal graphs achieving the $\ell$th least number of $C_4$ copies and advancing the supersaturation program for $C_4$ in dense regimes. The work has potential implications for related degenerate extremal problems and motivates further exploration of stability and supersaturation in geometric-inspired graph families.
Abstract
Extremal problems on the $4$-cycle $C_4$ played a heuristic important role in the development of extremal graph theory. A fundamental theorem of Füredi states that the Turán number $ex(q^2+q+1, C_4)\leq \frac12 q(q+1)^2$ holds for every $q\geq 14$, which matches with the classic construction of Erdős-R{é}nyi-Sós and Brown from finite geometry for prime powers $q$. Very recently, we obtained the first stability result on Füredi's theorem, by showing that for large even $q$, every $(q^2+q+1)$-vertex $C_4$-free graph with more than $\frac12 q(q+1)^2-0.2q$ edges must be a spanning subgraph of a unique polarity graph. Using new technical ideas in graph theory and finite geometry, we strengthen this by showing that the same conclusion remains true if the number of edges is lowered to $\frac12 q(q+1)^2-\frac12 q+o(q)$. Among other applications, this gives an immediate improvement on the upper bound of $ex(n,C_4)$ for infinitely many integers $n$. A longstanding conjecture of Erdős and Simonovits states that every $n$-vertex graph with $ex(n,C_4)+1$ edges contains at least $(1+o(1))\sqrt{n}$ 4-cycles. We proved an exact result and confirmed Erdős-Simonovits conjecture for infinitely many integers $n$. As the second main result of this paper, we further characterize all extremal graphs for which achieve the $\ell$th least number of copies of $C_4$ for any fixed positive integer $\ell$. This can be extended to more general settings and provides enhancements on the understanding of the supersaturation problem of $C_4$.