On a two-sided Turan problem

Given positive integers n, k, t, with 2 less than or equal to k less than or equal to n, and t < 2(k), let m(n, k, t) be the minimum size of a family F of nonempty subsets of [n] such that every k-set in [n] contains at least t sets from F, and every (k-1)-set in [n] contains at most t - 1 sets from F. Sloan et al. determined m( n, 3, 2) and Furedi et al. studied m(n, 4, t) for t = 2, 3. We consider m(n, 3, t) and m(n, 4, t) for all the remaining values of t and obtain their exact values except for k = 4 and t = 6, 7, 11, 12. For example, we prove that m(n, 4, 5) = ((n)(2)) - 17 for n greater than or equal to 160. The values of m(n, 4, t) for t = 7, 11, 12 are determined in terms of well-known (and open) Turan problems for graphs and hypergraphs. We also obtain bounds of m(n, 4, 6) that differ by absolute constants.