Let G be a simple graph that is properly edge-coloured with m colours and let \[\mathcal{M} = \{ {M_1},...,{M_m}\} \] be the set of m matchings induced by the colours in G. Suppose that \[m \leqslant n - {n^c}\], where \[c > 9/10\], and every matching in \[\mathcal{M}\] has size n. Then G contains a full rainbow matching, i.e. a matching that contains exactly one edge from Mi for each \[1 \leqslant i \leqslant m\]. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalization of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded multiplicity, and for hypergraphs.Finally, we provide counterexamples to several conjectures on full rainbow matchings made by Aharoni and Berger.

中文翻译：

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图和超图中的全彩虹匹配

让G是一个正确边缘着色的简单图米颜色并让\[\mathcal{M} = \{ {M_1},...,{M_m}\} \]是一组米由颜色引起的匹配G. 假设\[m \leqslant n - {n^c}\]， 在哪里\[c > 9/10\]，并且每个匹配\[\数学{M}\]有大小n. 然后G包含完整的彩虹匹配，IE恰好包含一条边的匹配米一世对于每个\[1 \leqslant i \leqslant m\]. 这回答了 Pokrovskiy 的一个未解决问题，并对 Aharoni 和 Berger 猜想的一个特殊情况的概括给出了肯定的回答。对于有界多重性边的多重图和超图，也发现了相关结果。最后，我们对 Aharoni 和 Berger 提出的关于全彩虹匹配的几个猜想提供了反例。