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 ) 5 n 2-1/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 Cn r−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.

We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs.