Non-uniform Turán-type problems

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 distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F , and every (k − 1)-subset of [n] contains at most t− 1 members of F . For fixed k and t, we determine the order of magnitude of m(n, k, t). We also consider related Turán numbers T≥r(n, k, t) and Tr(n, k, t), where T≥r(n, k, t) (Tr(n, k, t)) denotes the minimum size of a family F ⊂ ( [n] ≥r ) (F ⊂ ( [n] r ) ) such that every k-subset of [n] contains at least t members of F . We prove that T≥r(n, k, t) = (1+ o(1))Tr(n, k, t) for fixed r, k, t with t ≤ ( k r ) and n→∞.