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# Balanced supersaturation and Turan numbers in random graphs

Aug 2022

In a ground-breaking work utilizing the container method, Morris and Saxtonresolved a conjecture of Erd\H{o}s on the number of $C_{2\ell}$-free graphs on$n$ vertices and gave new bounds on the Turan number of $C_{2\ell}$ in theErd\H{o}s-Renyi random graph $G(n,p)$. A key ingredient of their work is theso-called balanced supersaturation property of even cycles of a given length.This motivated Morris and Saxton to make a broad conjecture of the existence ofsuch a property for all bipartite graphs. Roughly speaking, the conjecturestates that given a bipartite graph $H$ if an $n$-vertex graph $G$ has itsnumber of edges much larger than the Turan number $ex(n,H)$ then $G$ contains acollection of copies of $H$, in which no subset of edges of $G$ are coveredmore than some naturally expected number of times. In a subsequent breakthrough, Ferber, McKinley, and Samotij established aweaker version of the Morris-Saxton conjecture and applied it to derivefar-reaching results on the enumeration problem of $H$-free graphs. However, this weaker version seems insufficient for applications to the Turanproblem for random graphs. Building on these earlier works, in this paper, we essentially prove theconjecture of Morris and Saxton. We show that the conjecture holds when weimpose a very mild assumption about $H$, which is widely believed to hold forall bipartite graphs. In addition to retrieving the enumeration results ofFerber, McKinley, and Samotij, we also obtain some general upper bounds on theTuran number $ex(G(n,p), H)$ of a bipartite graph $H$ in the random graph$G(n,p)$, from which Morris and Saxton's result on $ex(G(n,p), C_{2\ell})$would also follow.

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