Rainbow matchings in non-uniform hypergraph systems

Let s≥ 1, r₁, r₂,, rₛ₊₁ be given positive integers and r=max₁, r₂,…, rₛ₊₁\. We prove that if F = ₁, …, Fₛ₊₁\ is a family of hypergraphs on a ground set of n vertices, n≥ 4r²(s+1), such that Fᵢ⊆ nrᵢ and |Fᵢ|>nrᵢ-n-srᵢ-n-s-rrᵢ-1+1, for all i ∈ [s+1], then F admits a rainbow matching of size $s+1$ or there exists W ∈ [n]s such that W is a vertex cover of Fᵢ, for all i ∈ [s+1]. This improves upon the previously known range n>2r³s to n>4r²(s+1). Moreover, for a given family F = ₁, …, Fₛ₊₁\ of hypergraphs Fᵢ⊆ nrᵢ, we give a best possible Ore-degree condition for Fᵢ's that guarantees the existence of a rainbow matching of size $s+1$ for F.