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   Abstract For and , let be a ‐partite ‐graph with parts each of size , where is sufficiently large. Assume that for each , every ‐set in lies in at least edges, and . We show that if , then contains a matching of size . In particular, contains a matching of size if each crossing ‐set lies in at least edges, or each crossing ‐set lies in at least edges and . This special case answers a question of Rödl and Ruciński and was independently obtained by Lu, Wang, and Yu. The proof of Lu, Wang, and Yu closely follows the approach of Han 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. 

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