Codegree Thresholds for Covering 3-Uniform Hypergraphs

Given two 3-uniform hypergraphs F and G = (V,E), we say that G has an F -covering if we can cover V with copies of F . The minimum codegree of G is the largest integer d such that every pair of vertices from V is contained in at least d triples from E. Define c2(n, F ) to be the largest minimum codegree among all n-vertex 3-graphs G that contain no F -covering. Determining c2(n, F ) is a natural problem intermediate (but distinct) from the well-studied Turán problems and tiling problems. In this paper, we determine c2(n,K4) (for n > 98) and the associated extremal configurations (for n > 998), where K4 denotes the complete 3-graph on 4 vertices. We also obtain bounds on c2(n, F ) which are apart by at most 2 in the cases where F is K− 4 (K4 with one edge removed), K − 5 , and the tight cycle C5 on 5 vertices.