{
"id": "1912.00986",
"version": "v1",
"published": "2019-12-02T18:24:46.000Z",
"updated": "2019-12-02T18:24:46.000Z",
"title": "Stability and supersaturation of $4$-cycles",
"authors": [
"Jialin He",
"Jie Ma",
"Tianchi Yang"
],
"categories": [
"math.CO"
],
"abstract": "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 $n$.",
"revisions": [
{
"version": "v1",
"updated": "2019-12-02T18:24:46.000Z"
}
],
"analyses": {
"keywords": [
"supersaturation",
"conjecture",
"extremal graph theory",
"heuristic important role",
"unique polarity graph"
],
"note": {
"typesetting": "TeX",
"pages": 0,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}