On rainbow matchings for hypergraphs

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 ≥ k2 ≥ . . . ≥ kt be positive integers. Huang, Loh and Sudakov asked for the maximum Πi=1|Ri| over all R = {R1, . . . ,Rt} such that each Ri is a collection of ki-subsets of [n] for which there does not exist a collection M of subsets of [n] such that |M | = t and |M ∩ Ri| = 1 for i ∈ [t] We show that for sufficiently large n with ∑t i=1 ki ≤ n(1 − (4k lnn/n) ), ∏t i=1 |Ri| ≤ ( n−1 k1−1 )( n−1 k2−1 ) ∏t i=3 ( n ki ) . This bound is tight.