Balanced supersaturation and Turan numbers in random graphs

TL;DR

This paper proves a conjecture on balanced supersaturation in bipartite graphs under mild conditions, leading to new enumeration results and bounds on Turán numbers in random graphs, extending understanding of $H$-free graphs.

Contribution

It confirms Morris and Saxton's conjecture for a broad class of bipartite graphs, enabling new bounds on Turán numbers in random graphs and advancing extremal graph theory.

Findings

Proves Morris and Saxton's conjecture under mild assumptions.

Establishes new upper bounds on Turán numbers in $G(n,p)$.

Derives enumeration results for $H$-free graphs.

Abstract

In a ground-breaking paper solving a conjecture of Erd\H{o}s on the number of $n$-vertex graphs not containing a given even cycle, Morris and Saxton \cite{MS} made a broad conjecture on so-called balanced supersaturation property of a bipartite graph $H$. Ferber, McKinley, and Samotij \cite{FMS} established a weaker version of this conjecture and applied it to derive far-reaching results on the enumeration problem of $H$-free graphs. In this paper, we show that Morris and Saxton's conjecture holds under a very mild assumption about $H$, which is widely believed to hold whenever $H$ contains a cycle. We then use our theorem to obtain enumeration results and general upper bounds on the Tur\'an number of a bipartite $H$ in the random graph $G(n,p)$, the latter being first of its kind.