On stability of rainbow matchings

We show that for any integer k≥ 1 there exists an integer t₀(k) such that, for integers t, k₁, … , kₜ₊₁, n with t> t₀(k) , max ₁, … , kₜ₊₁\≤ k , and n > 2k(t+1) , the following holds: If Fᵢ is a kᵢ -uniform hypergraph with vertex set $[n]$ and more than nkᵢ-n-tkᵢ - n-t-kkᵢ-1 + 1 edges for all i ∈ [t+1] , then either ₁,… , Fₜ₊₁\ admits a rainbow matching of size $t+1$ or there exists W∈ [n]t such that W intersects Fᵢ for all i∈ [t+1] . This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every t and n > 2k³t , generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings.