Set Systems without a Strong Simplex

Abstract

A d-simplex is a collection of $d+1$ sets such that every d of them have nonempty intersection and the intersection of all of them is empty. A strong d-simplex is a collection of $d+2$ sets $A,A_1,\dots,A_{d+1}$ such that $\{A_1,\dots,A_{d+1}\}$ is a d-simplex, while A contains an element of $\cap_{j\neq i}A_j$ for each i, $1\leq i\leq d+1$. Mubayi and Ramadurai [Combin. Probab. Comput., 18 (2009), pp. 441–454] conjectured that if $k\geq d+1\geq3$, $n>k(d+1)/d$, and $\mathcal{F}$ is a family of k-element subsets of an n-element set that contains no strong d-simplex, then $|\mathcal{F}|\leq{n-1\choose k-1}$ with equality only when $\mathcal{F}$ is a star. We prove their conjecture when $k\geq d+2$ and n is large. The case $k=d+1$ was solved in [M. Feng and X. J. Liu, Discrete Math., 310 (2010), pp. 1645–1647] and [Z. Füredi, private communication, St. Paul, MN, 2010]. Our result also yields a new proof of a result of Frankl and Füredi [J. Combin. Theory Ser. A, 45 (1987), pp. 226–262] when $k\geq d+2$ and n is large.