Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.08899 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915664192602112 |
|---|---|
| author | Joos, Felix Mattos, Letícia |
| author_facet | Joos, Felix Mattos, Letícia |
| contents | The Prague dimension of a graph $G$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph. Introduced in the 1970s by Nešetřil, Pultr, and Rödl -- and motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem -- determining the Prague dimension of a graph is a notoriously hard problem. In this paper, we show that for all $\varepsilon > 0$ and $p$ such that $ n^{-1+\varepsilon} \le p \le n^{-\varepsilon}$, with high probability the Prague dimension of $G_{n,p}$ is $Θ_{\varepsilon}(pn)$, which improves upon a recent result by Molnar, Rödl, Sales and Schacht.
Inspired by the work of Bennett and Bohman, our approach centres on analysing a random greedy process that builds an independent set of size $Ω(p^{-1}\log pn)$ by iteratively selecting vertices uniformly at random from the common non-neighbourhood of those already chosen. Using the differential equation method, we show that every non-edge is essentially equally likely to be covered by this process, which is key to establishing our bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Prague dimension of sparse random graphs Joos, Felix Mattos, Letícia Combinatorics The Prague dimension of a graph $G$ is defined as the minimum number of complete graphs whose direct product contains $G$ as an induced subgraph. Introduced in the 1970s by Nešetřil, Pultr, and Rödl -- and motivated by the work of Dushnik and Miller, as well as by the induced Ramsey theorem -- determining the Prague dimension of a graph is a notoriously hard problem. In this paper, we show that for all $\varepsilon > 0$ and $p$ such that $ n^{-1+\varepsilon} \le p \le n^{-\varepsilon}$, with high probability the Prague dimension of $G_{n,p}$ is $Θ_{\varepsilon}(pn)$, which improves upon a recent result by Molnar, Rödl, Sales and Schacht. Inspired by the work of Bennett and Bohman, our approach centres on analysing a random greedy process that builds an independent set of size $Ω(p^{-1}\log pn)$ by iteratively selecting vertices uniformly at random from the common non-neighbourhood of those already chosen. Using the differential equation method, we show that every non-edge is essentially equally likely to be covered by this process, which is key to establishing our bound. |
| title | On the Prague dimension of sparse random graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.08899 |