Universality for degenerate hypergraphs

A graph Γ is said to be universal for a class of graphs H if Γ contains a copy of every H ϵ H as a subgraph. The number of edges required for a host graph Γ to be universal for the class of D-degenerate graphs on n vertices has been shown to be O((log n)2/D(log log n)5n2-1/D). We generalise this result to r-uniform hypergraphs, showing the following. Given D, r ≥ 2 and n sufficiently large, there exists a constant C = C(D, r) such that there exists a graph with at most Cnr-1/D(log n)2/D(log log n)2r+1 edges, which is universal for the class of D-degenerate r-uniform hypergraphs on n vertices. This is tight up to the multiplicative constant and polylogarithmic term.