论文标题
SET系统中的单纯形群集的新结果
New results on simplex-clusters in set systems
论文作者
论文摘要
a $ d $ -simplex的定义为$ a_1，\ dots，a_ {d+1} $的大小$ k $ of $ $ [n] $的子集，使得它们的交集是空的，但是其中任何$ d $的交汇处都是无空的。此外，$ d $ - 集群是$ d+1 $的集合，具有空的交叉点和尺寸$ \ le 2k $的结合，以及$ d $ -simplex-cluster是这样的集合，既是$ d $ simplex又是$ d $ simplex和$ d $ cluster。 1974年的Erdős-Chvátal$ d $ -simplex的猜想指出，任何$ k $ -subsets的$ [n] $的家族包含no $ d $ -simplex必须大小不超过$ {n -1 \ select K-1} $。 2011年，凯瓦什（Keevash）和穆巴伊（Mubayi）通过假设对没有$ d $ simplex cluster的家庭的同样约束来扩展了这一猜想。在本文中，我们以所有$ 4 \ le D+1 \ le K $和$ n \ ge 2k-d+2 $解决Keevash和Mubayi的猜想，这又可以解决所有剩余的Erdős-Chvátal猜想的案例，除非$ n $很小，否则$ N $很小（I.E. $ n <2k-d+2 $）。
A $d$-simplex is defined to be a collection $A_1,\dots,A_{d+1}$ of subsets of size $k$ of $[n]$ such that the intersection of all of them is empty, but the intersection of any $d$ of them is non-empty. Furthermore, a $d$-cluster is a collection of $d+1$ such sets with empty intersection and union of size $\le 2k$, and a $d$-simplex-cluster is such a collection that is both a $d$-simplex and a $d$-cluster. The Erdős-Chvátal $d$-simplex Conjecture from 1974 states that any family of $k$-subsets of $[n]$ containing no $d$-simplex must be of size no greater than $ {n -1 \choose k-1}$. In 2011, Keevash and Mubayi extended this conjecture by hypothesizing that the same bound would hold for families containing no $d$-simplex-cluster. In this paper, we resolve Keevash and Mubayi's conjecture for all $4 \le d+1 \le k$ and $n \ge 2k-d+2$, which in turn resolves all remaining cases of the Erdős-Chvátal Conjecture except when $n$ is very small (i.e. $n < 2k-d+2$).
