On set systems without a simplex-cluster and the Junta method

A family $\{A_{0},\ldots,A_{d}\}$ of $k$-element subsets of $[n]=\{1,2,\ldots,n\}$ is called a simplex-cluster if $A_{0}\cap\cdots\cap A_{d}=\varnothing$, $|A_{0}\cup\cdots\cup A_{d}|\le2k$, and the intersection of any $d$ of the sets in $\{A_{0},\ldots,A_{d}\}$ is nonempty. In 2006, Keevash and Mubayi conjectured that for any $d+1\le k\le\frac{d}{d+1}n$, the largest family of $k$-element subsets of $[n]$ that does not contain a simplex-cluster is the family of all $k$-subsets that contain a given element. We prove the conjecture for all $k\ge\zeta n$ for an arbitrarily small $\zeta>0$, provided that $n\ge n_{0}(\zeta,d)$. We call a family $\{A_{0},\ldots,A_{d}\}$ of $k$-element subsets of $[n]$ a $(d,k,s)$-cluster if $A_{0}\cap\cdots\cap A_{d}=\varnothing$ and $|A_{0}\cup\cdots\cup A_{d}|\le s$. We also show that for any $\zeta n\le k\le\frac{d}{d+1}n$ the largest family of $k$-element subsets of $[n]$ that does not contain a $(d,k,(\frac{d+1}{d}+\zeta)k)$-cluster is again the family of all $k$-subsets that contain a given element, provided that $n\ge n_{0}(\zeta,d)$. Our proof is based on the junta method for extremal combinatorics initiated by Dinur and Friedgut and further developed by Ellis, Keller, and the author.