Full rainbow matchings in equivalence relations
Abstract
We show that if a multigraph G with maximum edge-multiplicity of at most n2 n, is edge-coloured by n colours such that each colour class is a disjoint union of cliques with at least 2n + o(n) vertices, then it has a full rainbow matching, that is, a matching where each colour appears exactly once. This asymptotically solves a question raised by Clemens, Ehrenm\"uller and Pokrovskiy, and is related to problems on algebras of sets studied by Grinblat in [Grinblat 2002].
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