Hypergraph universality via branching random walks

Abstract

Given a family of hypergraphs $\mathcal{H}$, we say that a hypergraph $\Gamma$ is $\mathcal{H}$-universal if it contains every $H \in \mathcal{H}$ as a subgraph. For $D, r \in \mathbb{N}$, we construct an $r$-uniform hypergraph with $\Theta\left(n^{r - r/D} \log^{r/D}(n)\right)$ edges which is universal for the family of all $r$-uniform hypergraphs with $n$ vertices and maximum degree at most $D$. This almost matches a trivial lower bound $\Omega(n^{r - r/D})$ coming from the number of such hypergraphs. On a high level, we follow the strategy of Alon and Capalbo used in the graph case, that is $r = 2$. The construction of $\Gamma$ is deterministic and based on a bespoke product of expanders, whereas showing that $\Gamma$ is universal is probabilistic. Two key new ingredients are a decomposition result for hypergraphs of bounded density, based on Edmond's matroid partitioning theorem, and a tail bound for branching random walks on expanders.