论文标题
关于当地的Turán问题
On local Turán problems
论文作者
论文摘要
自从其制定表达以来，Turán的超图问题一直是极端组合学中最具挑战性的开放问题之一。其中之一是以下内容：给定$ 3 $ - 均匀的超图$ \ MATHCAL {f} $上的$ n $ Vertices上，其中任何五个顶点至少跨越一个边缘，证明$ | \ Mathcal {f} | \ ge（1/4 -o（1））\ binom {n} {3} $。表明这种界限最好的结构就是$ \ binom {x} {3} \ cup \ binom {y} {3} {3} $，其中$ x $和$ y $ y $均匀分区the vertex set。该构造具有以下更通用的$（2P+1，p+1）$ - 属性：任何$ 2P+1 $ Vertices跨越$ P+1 $顶点的完整子Hypergraph。我们的主要结果之一表明，令人惊讶的是，对于所有$ p> 2 $（2p+1，p+1）$ - 属性意味着猜想的下限。
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 $|\mathcal{F}| \ge (1/4 -o(1))\binom{n}{3}$. The construction showing that this bound would be best possible is simply $\binom{X}{3} \cup \binom{Y}{3}$ 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.
