Abstract. Given positive integers a ≤ b ≤ c, let K a,b,c be the complete 3-partite 3-uniform hypergraph with three parts of sizes a, b, c. Let H be a 3-uniform hypergraph on n vertices where n is divisible by a + b + c. We asymptotically determine the minimum vertex degree of H that guarantees a perfect K a,b,ctiling, that is, a spanning subgraph of H consisting of vertex-disjoint copies of K a,b,c . This partially answers a question of Mycroft, who proved an analogous result with respect to codegree for r-uniform hypergraphs for all r ≥ 3. Our proof uses a lattice-based absorbing method, the concept of fractional tiling, and a recent result on shadows for 3-graphs.

For k ≥ 3 and ǫ > 0, let H be a k-partite k-graph with parts V 1 , . . . , V k each of size n, where n is sufficiently large. Assume that for each i ∈ [k], every (k − 1)-set in j∈[k]\{i} V i lies in at least a i edges, andIn particular, H contains a matching of size n − 1 if each crossing (k − 1)-set lies in at least ⌈n/k⌉ edges, or each crossing (k − 1)-set lies in at least ⌊n/k⌋ edges and n ≡ 1 mod k. 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 [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.

For k≥3 and ϵ>0, let H be a k‐partite k‐graph with parts V1,…,Vk each of size n, where n is sufficiently large. Assume that for each i∈[k], every (k−1)‐set in ∏j∈[k]\{i}Vj lies in at least ai edges, and a1≥a2≥⋯≥ak. We show that if a1,a2≥ϵn, then H contains a matching of size minfalse{n−1,∑i∈[k]aifalse}. In particular, H contains a matching of size n−1 if each crossing (k−1)‐set lies in at least ⌈n/k⌉ edges, or each crossing (k−1)‐set lies in at least ⌊n/k⌋ edges and n≡10.2emmod0.2emk. 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.