On set systems with a threshold property (Q856857)

For integers \(1 < t < k < n\) the set system \(\mathcal F\) on \([n]\) has the \((k,t)\)-threshold property if every \(k\)-subset of \([n]\) contains at least \(t\) sets from \(\mathcal F\) while every \((k-1)\)-subset of \([n]\) contains less than \(t\) sets from \(\mathcal F.\) The minimal cardinality of such a set system is denoted by \(m(n,k,t).\) This papers determines \(m(n,k,t)\) exactly for large enough \(n.\) It also shows that \(m(n,k,2)=(1+o(1))T_{(k-1)}(n,k,2)\) where the last item on the RHS is the generalized Turán number.