Abstract We show that for any integer k 1 there exists an integer t₀ (k) such that, for integers t, k₁, , kₓ+₁, n with t t₀ (k), \k₁, , kₓ+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 \F₁, , Fₓ+₁\ admits a rainbow matching of size t+1 or there exists W nt 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.
Lu et al. (Tue,) studied this question.
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