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arXiv · 2511.23341

Bounds for Hypergraph Universality

Peter Allen·Julia Böttcher·Jasmin Katz
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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.

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Peter Allen, Julia Böttcher, Jasmin Katz. 2025-11-28. Bounds for Hypergraph Universality. https://arxiv.org/abs/2511.23341

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