论文标题
随机图中的平衡超饱和和图兰数
Balanced supersaturation and Turan numbers in random graphs
论文作者
论文摘要
在一份开创性的纸张中，在不包含给定偶数周期的$ n $ vertex图上解决了erds的猜想，莫里斯和萨克斯顿\ cite {ms}对所谓的平衡均衡超饱和属性的广泛猜想。 Ferber，McKinley和Samotij \ Cite {FMS}建立了此猜想的较弱版本，并将其应用于$ H $ Free Graphs的枚举问题上的深远结果。 在本文中，我们表明，莫里斯和萨克斯顿的猜想在非常温和的假设下，大约$ h $，据信，每当$ h $包含一个周期时，据信它可以容纳。然后，我们使用定理在随机图$ g（n，p）$中获得两分$ h $的Turán数字上的枚举结果和一般上限，后者首先是同类。
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.
