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A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching.
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H. Hàn, Y. Person, and M. Schacht, On perfect matchings in uniform hypergraphs with large minimum vertex degree , SIAM J. Discrete Math. 23
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V. Rödl, A. Ruciński, and E. Szemerédi, Perfect matchings in large uniform hypergraphs with large minimum collective degree , J. Combin. Theory Ser. A 116
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F. Joos, J. Kim, D. Kühn, and D. Osthus, Optimal packings of bounded degree trees , J. Eur. Math. Soc. (to appear)
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R. Montgomery, A. Pokrovskiy, and B. Sudakov, Embedding rainbow trees with applications to graph labelling and decomposition , J. Eur. Math. Soc. (to appear)
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P. Condon, J. Kim, D. Kühn, and D. Osthus, A bandwidth theorem for approximate decompositions , Proc. Lond. Math. Soc. 118
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S. Ehard, S. Glock, and F. Joos, A rainbow blow-up lemma for almost optimally bounded edge-colourings , preprint (2019)
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J. Kim, D. Kühn, D. Osthus, and M. Tyomkyn, A blow-up lemma for approximate decompositions , Trans. Amer. Math. Soc. 371
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