On universal hypergraphs

A hypergraph $H$ is called universal for a family $\mathcal{F}$ of hypergraphs, if it contains every hypergraph $F\in \mathcal{F}$ as a copy. For the family of $r$-uniform hypergraphs with maximum vertex degree bounded by $\mathrm{\Delta }$ and at most $n$ vertices any universal hypergraph has to contain $\mathrm{\Omega }(n^{r-r\mathrm{/}\mathrm{\Delta }})$ many edges. We exploit constructions of Alon and Capalbo to obtain universal $r$-uniform hypergraphs with the optimal number of edges $O(n^{r-r\mathrm{/}\mathrm{\Delta }})$ when $r$ is even, $r\mid \mathrm{\Delta }$ or $\mathrm{\Delta }=2$. 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.