On Universal Hypergraphs

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