Bounds for Hypergraph Universality

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