Extremal hypergraphs and bounds for the Turán density of the 4-uniform K5K5

In this paper we find, for n≤16n≤16, the maximum number of edges in a 4-uniform hypergraph which does not have the complete 4-uniform hypergraph on five vertices, K54, as a subgraph. Equivalently, we find all optimal (n,n−4,n−5)(n,n−4,n−5) covering designs for n≤16n≤16.Using these results we find a new upper bound for the Turán density of K54. π(K54)≤17532380=0.73655…. Finally we make some notes on the structure of the extremal 4-graphs for this problem and the conjectured extremal family.