### Abstract

For positive integers s and k_{1}, k_{2}, ..., k_{s}, the van der Waerden number w (k_{1}, k_{2}, ..., k_{s} ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a k_{i}-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.

Original language | English (US) |
---|---|

Pages (from-to) | 1304-1309 |

Number of pages | 6 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 115 |

Issue number | 7 |

DOIs | |

State | Published - Oct 1 2008 |

### Fingerprint

### Keywords

- Arithmetic progressions
- Ramsey theory
- van der Waerden numbers

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series A*,

*115*(7), 1304-1309. https://doi.org/10.1016/j.jcta.2008.01.005

**Bounds on some van der Waerden numbers.** / Brown, Tom; Landman, Bruce M.; Robertson, Aaron.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series A*, vol. 115, no. 7, pp. 1304-1309. https://doi.org/10.1016/j.jcta.2008.01.005

}

TY - JOUR

T1 - Bounds on some van der Waerden numbers

AU - Brown, Tom

AU - Landman, Bruce M.

AU - Robertson, Aaron

PY - 2008/10/1

Y1 - 2008/10/1

N2 - For positive integers s and k1, k2, ..., ks, the van der Waerden number w (k1, k2, ..., ks ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.

AB - For positive integers s and k1, k2, ..., ks, the van der Waerden number w (k1, k2, ..., ks ; s) is the minimum integer n such that for every s-coloring of set {1, 2, ..., n}, with colors 1, 2, ..., s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w (k, m ; 2) for fixed m. We include a table of values of w (k, 3 ; 2) that are very close to this lower bound for m = 3. We also give a lower bound for w (k, k, ..., k ; s) that slightly improves previously-known bounds. Upper bounds for w (k, 4 ; 2) and w (4, 4, ..., 4 ; s) are also provided.

KW - Arithmetic progressions

KW - Ramsey theory

KW - van der Waerden numbers

UR - http://www.scopus.com/inward/record.url?scp=50649125454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50649125454&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2008.01.005

DO - 10.1016/j.jcta.2008.01.005

M3 - Article

VL - 115

SP - 1304

EP - 1309

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 7

ER -