Balanced supersaturation and Turan numbers in random graphs

Abstract

In a ground-breaking paper solving a conjecture of Erdős 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 Turán number of a bipartite $H$ in the random graph $G(n,p)$, the latter being first of its kind.

arXiv comment26 pages

References in corpus (7)

• Graphs with few paths of prescribed length between any two vertices
• Extremal results in random graphs
• The number of triple systems without even cycles
• Rational exponents near two
• Tree-degenerate graphs and nested dependent random choice
• The Turán number of blow-ups of trees
• Random Turán theorem for hypergraph cycles