Bounds for Hypergraph Universality

TL;DR

This paper extends bounds on the number of edges needed for hypergraphs to be universal for all D-degenerate r-uniform hypergraphs, generalizing known graph results with tight bounds up to polylogarithmic factors.

Contribution

It generalizes existing graph universality bounds to r-uniform hypergraphs, establishing near-optimal edge bounds for universality.

Findings

Established upper bounds for hypergraph universality with explicit edge counts.

Generalized graph universality results to hypergraphs with tight bounds.

Provided asymptotically optimal bounds up to polylogarithmic factors.

Abstract

A graph $\Gamma$ is said to be universal for a class of graphs $\mathcal{H}$ if $\Gamma$ contains a copy of every $H \in \mathcal{H}$ as a subgraph. The number of edges required for a host graph $\Gamma$ to be universal for the class of $D$-degenerate graphs on $n$ vertices has been shown to be $O(n^{2-1/D}(\log n)^{2/D}(\log\log n)^{5})$. 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 \[Cn^{r-1/D}(\log n)^{2/D}(\log\log n)^{2r+1}\] edges which is universal for the class of $D$-degenerate $r$-uniform hypergraphs on $n$ vertices. This is tight up to the polylogarithmic term.