### 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 |

Externally published | Yes |

### Keywords

- Arithmetic progressions
- Ramsey theory
- van der Waerden numbers

### ASJC Scopus subject areas

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

## Fingerprint Dive into the research topics of 'Bounds on some van der Waerden numbers'. Together they form a unique fingerprint.

## Cite this

Brown, T., Landman, B. M., & Robertson, A. (2008). Bounds on some van der Waerden numbers.

*Journal of Combinatorial Theory. Series A*,*115*(7), 1304-1309. https://doi.org/10.1016/j.jcta.2008.01.005