Bounds for Hypergraph Universality
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914173973168128 |
|---|---|
| author | Allen, Peter Böttcher, Julia Katz, Jasmin |
| author_facet | Allen, Peter Böttcher, Julia Katz, Jasmin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds for Hypergraph Universality Allen, Peter Böttcher, Julia Katz, Jasmin Combinatorics 05C65, 05D40 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. |
| title | Bounds for Hypergraph Universality |
| topic | Combinatorics 05C65, 05D40 |
| url | https://arxiv.org/abs/2511.23341 |