Extremal hypergraphs and bounds for the Turلn density of the 4-uniform

In this paper we find, for n ≤ 16 , the maximum number of edges in a 4-uniform hypergraph which does not have the complete 4-uniform hypergraph on five vertices, K 5 4 , as a subgraph. Equivalently, we find all optimal ( n , n − 4 , n − 5 ) covering designs for n ≤ 16 . these results we find a new upper bound for the Turán density of K 5 4 . π ( K 5 4 ) ≤ 1753 2380 = 0.73655 … . Finally we make some notes on the structure of the extremal 4-graphs for this problem and the conjectured extremal family.