Computing threshold functions by depth-3 threshold circuits with smaller thresholds of their gates
Information Processing Letters
On frequent sets of Boolean matrices
Annals of Mathematics and Artificial Intelligence
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Given positive integers n, k, t, with 2 ≤ k ≤ n, and t k, 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 T≥r(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 → ∞.