A Sharp Ramsey Theorem for Ordered Hypergraph Matchings
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913946982678528 |
|---|---|
| author | Sauermann, Lisa Zakharov, Dmitrii |
| author_facet | Sauermann, Lisa Zakharov, Dmitrii |
| contents | We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruciński. Namely, for any $r \ge 2$ and $n \ge 2$, we show that any collection $\mathcal H$ of $n$ pairwise disjoint subsets in $\mathbb Z$ of size $r$ contains a subcollection of size $\lfloor n^{1/(2^r-1)}/2\rfloor$ in which every pair of sets are in the same relative position with respect to the linear ordering on $\mathbb Z$. This improves previous bounds of Dudek-Grytczuk-Ruciński and of Anastos-Jin-Kwan-Sudakov and is sharp up to a factor of $2$. For large $r$, we even obtain such a subcollection of size $\lfloor (1-o(1))\cdot n^{1/(2^r-1)}\rfloor$, which is asymptotically tight (here, the $o(1)$-term tends to zero as $r \to \infty$, regardless of the value of $n$). Furthermore, we prove a multiparameter extension of this result where one wants to find a clique of prescribed size $m_P$ for each relative position pattern $P$. Our bound is sharp for all choices of parameters $m_P$, up to a constant factor depending on $r$ only. This answers questions of Anastos-Jin-Kwan-Sudakov and of Dudek-Grytczuk-Ruciński. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_04813 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Sharp Ramsey Theorem for Ordered Hypergraph Matchings Sauermann, Lisa Zakharov, Dmitrii Combinatorics We prove essentially sharp bounds for Ramsey numbers of ordered hypergraph matchings, inroduced recently by Dudek, Grytczuk, and Ruciński. Namely, for any $r \ge 2$ and $n \ge 2$, we show that any collection $\mathcal H$ of $n$ pairwise disjoint subsets in $\mathbb Z$ of size $r$ contains a subcollection of size $\lfloor n^{1/(2^r-1)}/2\rfloor$ in which every pair of sets are in the same relative position with respect to the linear ordering on $\mathbb Z$. This improves previous bounds of Dudek-Grytczuk-Ruciński and of Anastos-Jin-Kwan-Sudakov and is sharp up to a factor of $2$. For large $r$, we even obtain such a subcollection of size $\lfloor (1-o(1))\cdot n^{1/(2^r-1)}\rfloor$, which is asymptotically tight (here, the $o(1)$-term tends to zero as $r \to \infty$, regardless of the value of $n$). Furthermore, we prove a multiparameter extension of this result where one wants to find a clique of prescribed size $m_P$ for each relative position pattern $P$. Our bound is sharp for all choices of parameters $m_P$, up to a constant factor depending on $r$ only. This answers questions of Anastos-Jin-Kwan-Sudakov and of Dudek-Grytczuk-Ruciński. |
| title | A Sharp Ramsey Theorem for Ordered Hypergraph Matchings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2309.04813 |