Improved Ramsey-type theorems for Fibonacci numbers and other sequences
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2022
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911137316995072 |
|---|---|
| author | Wesley, William J. |
| author_facet | Wesley, William J. |
| contents | Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression. We consider two variants of van der Waerden numbers: the numbers $n = n(AP_D,k;r)$, the smallest value where every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression with common difference in $D$, and the numbers $n = Δ(D,k;r)$, the smallest value $n$ where every $r$-coloring of $\{1,\dots,n\}$ contains a sequence $x_1 < \dots < x_k$ where the differences between consecutive terms are members of $D$. We study the case when $D$ is set of Fibonacci numbers $F$ and give improved bounds for the largest $r$ where $n(AP_F,k;r)$ and $Δ(F,k;r)$ exist for all $k$. Moreover, we give some computational data on $Δ(D,k;r)$ for other sets $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05167 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Improved Ramsey-type theorems for Fibonacci numbers and other sequences Wesley, William J. Combinatorics Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression. We consider two variants of van der Waerden numbers: the numbers $n = n(AP_D,k;r)$, the smallest value where every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term arithmetic progression with common difference in $D$, and the numbers $n = Δ(D,k;r)$, the smallest value $n$ where every $r$-coloring of $\{1,\dots,n\}$ contains a sequence $x_1 < \dots < x_k$ where the differences between consecutive terms are members of $D$. We study the case when $D$ is set of Fibonacci numbers $F$ and give improved bounds for the largest $r$ where $n(AP_F,k;r)$ and $Δ(F,k;r)$ exist for all $k$. Moreover, we give some computational data on $Δ(D,k;r)$ for other sets $D$. |
| title | Improved Ramsey-type theorems for Fibonacci numbers and other sequences |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2211.05167 |