Highly Influential

For any posotive integer m, let [m] := {1, . . . ,m}. Let n, k, t be positive integers. Aharoni and Howard conjectured that if, for i ∈ [t], Fi ⊂ [n] := {(a1, . . . , ak) : aj ∈ [n] for j ∈ [k]} and |Fi| > (t−1)n, then there exist M ⊆ [n] such that |M | = t and |M ∩ Fi| = 1 for i ∈ [t] We show that this conjecture holds when n ≥ 3(k − 1)(t− 1). Let n, t, k1… (More)