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).
- Extremal problem
- Set system
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics