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arXiv · 2208.10572

Balanced supersaturation and Turan numbers in random graphs

Tao Jiang·Sean Longbrake
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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.

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Tao Jiang, Sean Longbrake. 2022-08-22. Balanced supersaturation and Turan numbers in random graphs. https://doi.org/10.19086/aic.2024.3

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