Exact Minimum Codegree Threshold for K − 4 -Factors

Source abstract

Given hypergraphs F and H , an F -factor in H is a set of vertex-disjoint copies of F which cover all the vertices in H . Let K − 4 denote the 3-uniform hypergraph with four vertices and three edges. We show that for sufficiently large n ∈ 4ℕ, every 3-uniform hypergraph H on n vertices with minimum codegree at least n /2−1 contains a K − 4 -factor. Our bound on the minimum codegree here is best possible. It resolves a conjecture of Lo and Markström [15] for large hypergraphs, who earlier proved an asymptotically exact version of this result. Our proof makes use of the absorbing method as well as a result of Keevash and Mycroft [11] concerning almost perfect matchings in hypergraphs.