On universal hypergraphs

TL;DR

This paper studies the construction of universal hypergraphs that contain all hypergraphs of a certain family, achieving optimal edge counts in specific cases, and generalizes known results from graphs to hypergraphs.

Contribution

It provides constructions of universal hypergraphs with optimal edge counts for certain parameters and extends existing graph results to hypergraphs.

Findings

Constructed universal hypergraphs with optimal edge counts for even r or specific Δ.

Extended graph universality results to hypergraphs.

Achieved bounds matching theoretical lower limits.

Abstract

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