3 problems

Fix integers $n\ge q>r\ge 2$. A partial $(n,q,r)$-Steiner system is a collection of $q$-subsets of an $n$-element set such that every $r$-subset lies in at most one $q$-set; an…

Let $X$ be a set of size $n$. A Steiner triple system of order $n$ is a collection of $3$-subsets of $X$ in which every $2$-subset of $X$ lies in exactly one triple. Call $n$ admis…

Let $K^{(r)}_{n}$ be the complete $r$-graph on $n$ vertices. A block design with given parameters is a design whose underlying hypergraph is $K^{(r)}_{n}$; such a design can exist…