Tournaments, 4-uniform hypergraphs, and an exact extremal result

We consider 4-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of 5 vertices spans either 0 or exactly 2 hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a ‘switching’ operation on tournaments that preserves hypergraphs arising from this construction.