集合系统中单纯形簇的新结果
New results on simplex-clusters in set systems
摘要：
$d$ -单纯形被定义为$[n]$的大小为$k$的子集的集合$A_1,\dots,A_{d+1}$，使得它们所有的交集为空，但是它们中的任何$d$的交集为非空。此外，$d$ -簇是具有大小为$\le 2k$的空交集和并集的$d+1$这样的集合的集合，并且$d$ -单纯形-簇是既是$d$ -单纯形又是$d$ -簇的这样的集合。1974年的Erdens-Chvátal $d$ -单纯形猜想指出，任何不含$d$ -单纯形的$k$ -$[n]$子集族的大小必须不大于$ {n -1 \choose k-1}$。在2011年，Keevash和Mubayi扩展了这个猜想，假设相同的界限也适用于不包含$d$ -单纯-簇的族。在本文中，我们解决了Keevash和Mubayi的猜想，所有的$4 \le d+1 \le k$和$n \ge 2k-d+2$，这反过来解决了所有剩余的情况下，Erdens-Chvátal猜想，除了当$n$是非常小的（即$n < 2k-d+2$）。
英文摘要：
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. $n < 2k-d+2$).
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