Bounds for Hypergraph Universality

TL;DR

The paper extends universality results from graphs to D-degenerate r-uniform hypergraphs, establishing the existence of an H-universal r-graph on n vertices with at most C n^{r-1/D} (log n)^{2/D} (log log n)^{2r+1} edges for large n, and proving a matching lower bound up to polylog factors. The approach combines a random block model Γ(r,n,D) with a degeneracy-aware embedding strategy, underpinned by a pseudorandomness lemma that guarantees plentiful embedding candidates. This work tightens the understanding of hypergraph universality and highlights directions for spanning universality and bounded-density extensions. The results significantly bridge the gap between graph and hypergraph universality theories, offering a robust construction and analysis toolkit for r-uniform hypergraphs.

Abstract

A graph $Γ$ is said to be universal for a class of graphs $\mathcal{H}$ if $Γ$ contains a copy of every $H \in \mathcal{H}$ as a subgraph. The number of edges required for a host graph $Γ$ 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.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.2: Stirling's Approximation
• Theorem 2.3: Chernoff bound Chernoff_one_sided
• proof : Proof of Theorem \ref{['thm:lower_bd']}
• Definition 4.1: Random block model, $\Gamma(r, n, D)$
• Proposition 4.2: Properties of the Model
• proof
• Definition 4.3: Embedding Strategy
• Definition 4.4: Well-behaved multi-set