On a Two-Sided Turan Problem

Given positive integers n, k, t, with 2 ≤ k ≤ n, and t < 2, 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 [11] determined m(n, 3, 2) and Füredi et. al [6] 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 ≥ 160. The values of m(n, 4, t) for t = 7, 11, 12 are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of m(n, 4, 6) that differ by absolute constants.