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. Furthemore, 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-Chvatal $d$-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 $4 \le d+1 \le k \le n/2$, which in turn resolves all remaining cases of the Erdős-Chvatal Conjecture except when $n$ is very small (i.e. $n < 2k$).

中文翻译：

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集合系统中单纯形簇的新结果

$d$-simplex 定义为 $[n]$ 大小为 $k$ 的子集的集合 $A_1​​,\dots,A_{d+1}$，使得它们的交集为空，但是它们中任何 $d$ 的交集都是非空的。此外，$d$-cluster 是 $d+1$ 这样的集合的集合，这些集合具有空交集和大小为 $\le 2k$ 的并集，而 $d$-simplex-cluster 是这样一个集合，它既是 $ d$-simplex 和 $d$-cluster。1974 年的 Erdős-Chvatal $d$-simplex 猜想指出，任何不包含 $d$-simplex 的 $[n]$ 的 $k$-子集的大小必须不大于 ${n -1 \choose k -1}$。2011 年，Keevash 和 Mubayi 通过假设相同的界限将适用于不包含 $d$-simplex-cluster 的家庭，从而扩展了这一猜想。在本文中，我们解决了所有 $4 \le d+1 \le k \le n/2$ 的 Keevash 和 Mubayi 猜想，