Since its formulation, Turán's hypergraph problems have been among the most challenging open problems in extremal combinatorics. One of them is the following: given a 3-uniform hypergraph $\mathcal{F}$ on n vertices in which any five vertices span at least one edge, prove that $|\mathcal{F}|\ge (1/4-o(1))\left(\begin{array}{c}n\\ 3\end{array}\right)$. The construction showing that this bound would be best possible is simply $\left(\begin{array}{c}X\\ 3\end{array}\right)\cup \left(\begin{array}{c}Y\\ 3\end{array}\right)$ where X and Y evenly partition the vertex set. This construction has the following more general $(2p+1,p+1)$-property: any set of $2p+1$ vertices spans a complete sub-hypergraph on $p+1$ vertices. One of our main results says that, quite surprisingly, for all $p>2$ the $(2p+1,p+1)$-property implies the conjectured lower bound.

自从提出以来，图兰的超图问题一直是极值组合学中最具挑战性的开放性问题。其中之一是：给定3一致的超图$\mathcal{F}$在任意五个顶点至少跨越一条边的n个顶点上，证明$|\mathcal{F}|\ge （\mathrm{1个}/4-Ø（\mathrm{1个}））（\begin{array}{c}ñ\\ 3\end{array}）$。显示此界限最可能的结构只是$（\begin{array}{c}X\\ 3\end{array}）\cup （\begin{array}{c}ÿ\\ 3\end{array}）$其中X和Y均匀分割顶点集。这个构造有以下更一般的$（2p+\mathrm{1个}，p+\mathrm{1个}）$-属性：任何一组 $2p+\mathrm{1个}$ 顶点跨越一个完整的子超图 $p+\mathrm{1个}$顶点。我们的主要结果之一表明，对于所有人$p>2$ 的 $（2p+\mathrm{1个}，p+\mathrm{1个}）$-属性暗示推测的下界。