论文标题
没有非平凡相交子图的超图的注释
A note on hypergraphs without non-trivial intersecting subgraphs
论文作者
论文摘要
如果每两个边缘都有非空的交叉点，则HyperGraph $ \ Mathcal {F} $是非平凡的相交，但是在$ \ Mathcal {f} $的所有边缘中都没有顶点。 Mubayi和Verstraëte表明，每$ k \ ge d+1 \ ge 3 $和$ n \ ge（d+1）n/d n/d n/d n/d n/d n/d n/d n/d n/d n $ k $ -graph $ \ mathcal {h} $ on $ n $ vertices上的$ n $ vertices上，没有非物质相互交叉的$ d+d $ d $ d $ d $ d $ $ \ binom的n $ d+binom} k-1 {k-1 {他们猜想所有$ d \ ge k \ ge 4 $和足够大的$ n $都有相同的结论。我们通过证明更强有力的声明来确认他们的猜想。 他们还推测，对于$ m \ ge 4 $，并且足够大的$ n $最大尺寸的最大尺寸是$ n $ vertices上的$ 3 $绘图，而没有某些施气纳系统可实现$ 3M+1 $的非平凡的相互作用的子图。我们提供了一个构造，其边缘更多，表明它们的猜想一般不是事实。
A hypergraph $\mathcal{F}$ is non-trivial intersecting if every two edges in it have a nonempty intersection but no vertex is contained in all edges of $\mathcal{F}$. Mubayi and Verstraëte showed that for every $k \ge d+1 \ge 3$ and $n \ge (d+1)n/d$ every $k$-graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d \ge k \ge 4$ and sufficiently large $n$. We confirm their conjecture by proving a stronger statement. They also conjectured that for $m \ge 4$ and sufficiently large $n$ the maximum size of a $3$-graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner systems. We give a construction with more edges showing that their conjecture is not true in general.
