For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem in extremal combinatorics. While for $k=2$ this problem is well understood, due to the work of Turán and of Erdős and Stone, only very little is known for $k$-uniform hypergraphs for $k>2$. We focus on the case when $F$ is a $k$-uniform hypergraph with three edges on $k+1$ vertices. Already this very innocent (and maybe somewhat particular looking) problem is still wide open even for $k=3$.
We consider a variant of the problem where the large hypergraph $H$ enjoys additional hereditary density conditions. Questions of this type were suggested by Erd\H os and Sós about 30 years ago. We show that every $k$-uniform hypergraph $H$ with density $>2^{1-k}$ with respect to every large collections of $k$-cliques induced by sets of $(k-2)$-tuples contains a copy of $F$. The required density $2^{1-k}$ is best possible as higher order tournament constructions show.
Our result can be viewed as a common generalisation of the first extremal result in graph theory due to Mantel (when $k=2$ and the hereditary density condition reduces to a normal density condition) and a recent result of Glebov, Král', and Volec (when $k=3$ and large subsets of vertices of $H$ induce a subhypergraph of density $>1/4$). Our proof for arbitrary $k\geq 2$ utilises the regularity method for hypergraphs.
极值图论中的一个经典结果是Mantel定理，它指出$K_n$的每个最大无三角形子图都是二部图。Mantel定理的一个稀疏版本是，对于足够大的$p$，$G(n,p)$的每个最大无三角形子图是w. h. p.二部图。最近，DeMarco和Kahn证明了对于$p > K \sqrt{\log n/n}$，对于某个常数$K$，这个界是最好的。本文研究了随机超图中这类极值问题。用$F_5$表示顶点集为{a，b，c，d，e}，边集为{abc，ade，bde}的3-一致超图，有时也称为广义三角形。极值超图理论中的第一个极值结果是由Frankl和Füredi证明的，当n>3000时，n个顶点上不包含$F_5$拷贝的最大3-一致超图是三部的。一个自然的问题是什么p是$G^3(n,p)$ w. h. p.三分图的每个最大$F_5$ -自由子超图。我们证明了这对$p>K\log n/n$对某个常数K成立，而对$p=0.1\sqrt{\log n}/n$不成立。
