The Keevash--Mubayi simplex-cluster conjecture

Abstract

A $d$-simplex-cluster is a collection of $d+1$ distinct $k$-element sets with empty total intersection, nonempty intersection for every $d$ members, and union of size at most $2k$. We prove the simplex-cluster conjecture of Keevash and Mubayi, a common strengthening of the Erdős--Chvátal simplex conjecture and Mubayi's cluster conjecture. More precisely, for integers $k>d\ge2$ and $n\ge k(d+1)/d$, every family of $k$-element subsets of an $n$-element set containing no $d$-simplex-cluster has at most $\binom{n-1}{k-1}$ members. Equality holds if and only if the family consists of all $k$-element subsets containing a fixed point.