Supersaturation of $C_4$: from Zarankiewicz towards Erd\H{o}s-Simonovits-Sidorenko

TL;DR

This paper investigates the minimum number of specific bipartite subgraphs in dense bipartite graphs, providing exact and asymptotic results for certain cases, and explores connections to combinatorial design theory.

Contribution

It offers new exact and asymptotic bounds for the number of $K_{2,t}$ and $C_4$ subgraphs in bipartite graphs, advancing understanding of supersaturation and related conjectures.

Findings

Exact results for $C_4$ when $m$ and Zarankiewicz number differ by at most $n

Asymptotically sharp bounds for $K_{2,t}$ when difference is $Cn\sqrt{n}$

Connections established between supersaturation, covering, and packing block designs

Abstract

For a positive integer $n$, a graph $F$ and a bipartite graph $G\subseteq K_{n,n}$ let ${F(n+n, G)}$ denote the number of copies of $F$ in $G$, and let $F(n+n, m)$ denote the minimum number of copies of $F$ in all graphs $G\subseteq K_{n,n}$ with $m$ edges. The study of such a function is the subject of theorems of supersaturated graphs and closely related to the Sidorenko-Erd\H{o}s-Simonovits conjecture as well. In the present paper we investigate the case when $F= K_{2,t}$ and in particular the quadrilateral graph case. For $F=C_4$, we obtain exact results if $m$ and the corresponding Zarankiewicz number differ by at most $n$, by a finite geometric construction of almost difference sets. $F= K_{2,t}$ if $m$ and the corresponding Zarankiewicz number differs by $Cn\sqrt{n}$ we prove asymptotically sharp results. We also study stability questions and point out the connections to covering and packing block designs.