2 problems

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote its power set, and let $\beta(\mathcal{F})$ denote the sturdiness of a family $\mathcal{F}\subset 2^{[n]}$. A family is $(2s+1)$-unio…

Let $p(n,k,d)$ denote the maximum size of a family $\mathcal{A}\subseteq 2^{[n]}$ such that the union of every $k$ members of $\mathcal{A}$ has size at most $d$, where $n,k,d>0$ an…