On set systems with a threshold property

For n, k and t such that 1 < t < k < n, a set F of subsets of [n] has the (k, t)threshold property if every k-subset of [n] contains at least t sets from F and every (k − 1)-subset of [n] contains less than t sets from F . The minimal number of sets in a set system with this property is denoted by m(n, k, t). In this paper we determine m(n, 4, 3) exactly for n sufficiently large, and we show that m(n, k, 2) is asymptotically equal to the generalized Turán number Tk−1(n, k, 2). Preprint submitted to Elsevier Science 21 December 2005