Approximate Steiner $(r-1,r,n)$-systems without $3$ blocks on $r+2$ points

Abstract: For a family ${\mathcal F}$ of $r$-graphs, let $\mathrm{ex}(n,{\mathcal F})$ denote the maximum number of edges in an ${\mathcal F}$-free $r$-graph on $n$ vertices. Let ${\mathcal F}_r(v,e)$ denote the family of all $r$-graphs with $e$ edges and at most $v$ vertices. We prove that $$\mathrm{ex}(n,{\mathcal F}_r(r+1,2) \cup {\mathcal F}_r(r+2,3)) = (\frac{1}{r} - o(1)) \binom{n}{r-1}$$.