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 $\mathrm{ex}(n,{\mathcal{F}}_r(r+1,2)\cup {\mathcal{F}}_r(r+2,3))=(\frac{1}{r}-o(1))\left(\genfrac{}{}{0ex}{}{n}{r-1}\right)$.

Final version with the changes suggested by the referees. To appear in "Journal of Combinatorial Designs"