Families of Sets with Intersecting Clusters

A family of $k$-subsets $A_1, A_2, ..., A_d$ on $[n]=\{1,2,..., n\}$ is called a $(d, c)$-cluster if the union $A_1\cup A_2 \cup ... \cup A_d$ contains at most $ck$ elements with $c<d$. Let $\mathcal{F}$ be a family of $k$-subsets of an $n$-element set. We show that for $k \geq 2$ and $n \geq k+2$, if every $(k, 2)$-cluster of $\mathcal{F}$ is intersecting, then $\mathcal{F}$ contains no $(k-1)$-dimensional simplices. This leads to an affirmative answer to Mubayi's conjecture for $d=k$ based on Chv\'atal's simplex theorem. We also show that for any $d$ satisfying $3 \leq d \leq k$ and $n \geq \frac{dk}{d-1}$, if every $(d, {d+1\over 2})$-cluster is intersecting, then $|\mathcal{F}|\leq {{n-1} \choose {k-1}}$ with equality only when $\mathcal{F}$ is a complete star. This result is an extension of both Frankl's theorem and Mubayi's theorem.