3 problems

Let $\mathcal{F}\subseteq\binom{[n]}{k}$ be $3$-cluster-free, meaning it contains no three distinct $k$-sets with empty common intersection and union of size at most $2k$. A comple…

Let $\mathcal{F}\subseteq\binom{[n]}{k}$. A $3$-cluster is a collection of three distinct $k$-sets with empty common intersection and union of size at most $2k$; $\mathcal{F}$ is…

Let $[n]$ denote the set $\{1,\ldots,n\}$, and let $\binom{[n]}{k}$ be the collection of all $k$-sets of $[n]$. A family is $d$-cluster-free if it contains no collection of $d$ dif…