Matchings in k-partite k-uniform hypergraphs

For (Formula presented.) and (Formula presented.), let (Formula presented.) be a (Formula presented.) -partite (Formula presented.) -graph with parts (Formula presented.) each of size (Formula presented.), where (Formula presented.) is sufficiently large. Assume that for each (Formula presented.), every (Formula presented.) -set in (Formula presented.) lies in at least (Formula presented.) edges, and (Formula presented.). We show that if (Formula presented.), then (Formula presented.) contains a matching of size (Formula presented.). In particular, (Formula presented.) contains a matching of size (Formula presented.) if each crossing (Formula presented.) -set lies in at least (Formula presented.) edges, or each crossing (Formula presented.) -set lies in at least (Formula presented.) edges and (Formula presented.). 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.