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 $Δ$ and at most $n$ vertices any universal hypergraph has to contain $Ω(n^{r-r/Δ})$ many edges. We exploit constructions of Alon and Capalbo to obtain universal $r$-uniform hypergraphs with the optimal number of edges $O(n^{r-r/Δ})$ when $r$ is even, $r \mid Δ$ or $Δ=2$. 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.

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