In this short note we prove that every tournament contains the k-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when k=2, showing that there is always a square of a directed path of length , which is best possible.

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A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that "every edge coloured graph of a certain type contains a rainbow matching using every colour". In this paper we introduce a versatile "sampling trick", which allows us to obtain short proofs of old results as well as to solve asymptotically some well known conjectures.• We give a simple proof of Pokrovskiy's asymptotic version of the Aharoni-Berger conjecture with greatly improved error term.• We give the first asymptotic proof of the "non-bipartite" Aharoni-Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky and Zerbib.• We give a very short asymptotic proof of Grinblat's conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat's problem as a function of edge multiplicity of the corresponding multigraph.• We give the first asymptotic proof of a 30 year old conjecture of Alspach.

We show that if a multigraph G with maximum edge-multiplicity of at most √ n log 2 n , is edgecoloured 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, Ehrenmüller and Pokrovskiy, and is related to problems on algebras of sets studied by Grinblat in [Grinblat 2002].

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