On Universal Hypergraphs

A hypergraph H is called universal for a family F of hypergraphs, if it contains every hypergraph F is an element of F as a copy. For the family of r-uniform hypergraphs with maximum vertex degree bounded by Delta and at most n vertices any universal hypergraph has to contain Omega(n(r-r)/(Delta)) many edges. We exploit constructions of Alon and Capalbo to obtain universal r-uniform hypergraphs with the optimal number of edges O(n(r-r)/(Delta)) when r is even, r vertical bar Delta or Delta = 2. Further we generalize the result of Alon and Asodi about optimal universal graphs for the family of graphs with at most rn edges and no isolated vertices to hypergraphs.