For $k\ge 3$ and $\epsilon>0$, let $H$ be a $k$-partite $k$-graph with parts $V_1,\dots, V_k$ each of size $n$, where $n$ is sufficiently large. Assume that for each $i\in [k]$, every $(k-1)$-set in $\prod_{j\in [k]\setminus \{i\}} V_i$ lies in at least $a_i$ edges, and $a_1\ge a_2\ge \cdots \ge a_k$. We show that if $a_1, a_2\ge \epsilon n$, then $H$ contains a matching of size $\min\{n-1, \sum_{i\in [k]}a_i\}$. In particular, $H$ contains a matching of size $n-1$ if each crossing $(k-1)$-set lies in at least $\lceil n/k \rceil$ edges, or each crossing $(k-1)$-set lies in at least $\lfloor n/k \rfloor$ edges and $n\equiv 1\bmod k$. This special case answers a question of Rodl and Rucinski and was independently obtained by Lu, Wang, and Yu. The proof of Lu, Wang, and Yu closely follows the approach of Han [Combin. Probab. Comput. 24 (2015), 723--732] by using the absorbing method and considering an extremal case. In contrast, our result is more general and its proof is thus more involved: it uses a more complex absorbing method and deals with two extremal cases.

中文翻译：

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k-partite k-uniform hypergraphs 中的匹配

对于 $k\ge 3$ 和 $\epsilon>0$，令 $H$ 是一个 $k$-partite $k$-graph，其中部分 $V_1,\dots, V_k$ 的大小分别为 $n$，其中 $ n$ 足够大。假设对于每个 $i\in [k]$，$\prod_{j\in [k]\setminus \{i\}} 中的每个 $(k-1)$-set V_i$ 至少位于 $a_i $ 边和 $a_1\ge a_2\ge \cdots \ge a_k$。我们证明如果 $a_1, a_2\ge \epsilon n$, 那么 $H$ 包含大小为 $\min\{n-1, \sum_{i\in [k]}a_i\}$ 的匹配。特别地，如果每个交叉 $(k-1)$-set 位于至少 $\lceil n/k \rceil$ 边，或者每个交叉 $(k- 1)$-set 至少位于 $\lfloor n/k \rfloor$ 边和 $n\equiv 1\bmod k$ 中。这个特例回答了 Rodl 和 Rucinski 的问题，由 Lu、Wang 和 Yu 独立获得。鲁、王、禹的证明与韩[结合．可能。计算。24 (2015), 723--732]通过使用吸收方法并考虑极值情况。相比之下，我们的结果更一般，因此其证明也更复杂：它使用更复杂的吸收方法并处理两个极值情况。