On the Independence Numbers of the Cyclic Van der Waerden Hypergraphs
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908544724369408 |
|---|---|
| author | Liber, Benjamin |
| author_facet | Liber, Benjamin |
| contents | Building upon the work of Berglund (2018), we establish a method for constructing subsets $B \subseteq \mathbb{Z}_{mk}$ such that $B$ does not contain any $k$-term cyclic arithmetic progressions mod $mk$, where $m,k \in \mathbb{Z}^+$ with $k \geq 3$. This construction thereby provides concrete lower bounds for the maximum size of such subsets. Additionally, it allows us to tightly bound specific chromatic numbers $χ(mk,k)$ of $\mathbb{Z}_{mk}$ and helps increase the lower bounds of certain cyclic Van der Waerden numbers $W_{c}(k,r)$, originally introduced by Burkert and Johnson (2011) as a way of bounding the standard Van der Waerden numbers $W(k,r)$ from below for $r \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Independence Numbers of the Cyclic Van der Waerden Hypergraphs Liber, Benjamin Combinatorics 05D10 (Primary) 11B25, 05C15 (Secondary) Building upon the work of Berglund (2018), we establish a method for constructing subsets $B \subseteq \mathbb{Z}_{mk}$ such that $B$ does not contain any $k$-term cyclic arithmetic progressions mod $mk$, where $m,k \in \mathbb{Z}^+$ with $k \geq 3$. This construction thereby provides concrete lower bounds for the maximum size of such subsets. Additionally, it allows us to tightly bound specific chromatic numbers $χ(mk,k)$ of $\mathbb{Z}_{mk}$ and helps increase the lower bounds of certain cyclic Van der Waerden numbers $W_{c}(k,r)$, originally introduced by Burkert and Johnson (2011) as a way of bounding the standard Van der Waerden numbers $W(k,r)$ from below for $r \geq 2$. |
| title | On the Independence Numbers of the Cyclic Van der Waerden Hypergraphs |
| topic | Combinatorics 05D10 (Primary) 11B25, 05C15 (Secondary) |
| url | https://arxiv.org/abs/2509.07926 |