### Abstract

For integers b≥0 and c≥ 1, define f_{c}(b) to be the least positive integer n such that for every 2-coloring of [1, n] there is a monochromatic sequence of the form {x, x + d, x + 2d + b} where x and d are positive integers with d ≥c. Bialostocki, Lefmann, and Meerdink showed that for b even, 2b + 10≤f⊥(b)≤13/2b + 1, where the lower bound holds for b≥10. We find upper and lower bounds for the more general function f_{c}(b) which, for c = 1, improve the aforementioned upper bound to ⌈9b/4⌉ + 9, and give the same lower bound of 2b + 10. Results about f_{c}(b) are used to find analogous results on a slightly different generalization of f⊥(b).

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

Pages (from-to) | 137-147 |

Number of pages | 11 |

Journal | Discrete Mathematics |

Volume | 207 |

Issue number | 1-3 |

DOIs | |

State | Published - Sep 28 1999 |

Externally published | Yes |

### Fingerprint

### Keywords

- Monochromatic sequences
- Van der Waerden numbers

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

**On some generalizations of the van der Waerden number w(3).** / Landman, Bruce M.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 207, no. 1-3, pp. 137-147. https://doi.org/10.1016/S0012-365X(99)00137-5

}

TY - JOUR

T1 - On some generalizations of the van der Waerden number w(3)

AU - Landman, Bruce M.

PY - 1999/9/28

Y1 - 1999/9/28

N2 - For integers b≥0 and c≥ 1, define fc(b) to be the least positive integer n such that for every 2-coloring of [1, n] there is a monochromatic sequence of the form {x, x + d, x + 2d + b} where x and d are positive integers with d ≥c. Bialostocki, Lefmann, and Meerdink showed that for b even, 2b + 10≤f⊥(b)≤13/2b + 1, where the lower bound holds for b≥10. We find upper and lower bounds for the more general function fc(b) which, for c = 1, improve the aforementioned upper bound to ⌈9b/4⌉ + 9, and give the same lower bound of 2b + 10. Results about fc(b) are used to find analogous results on a slightly different generalization of f⊥(b).

AB - For integers b≥0 and c≥ 1, define fc(b) to be the least positive integer n such that for every 2-coloring of [1, n] there is a monochromatic sequence of the form {x, x + d, x + 2d + b} where x and d are positive integers with d ≥c. Bialostocki, Lefmann, and Meerdink showed that for b even, 2b + 10≤f⊥(b)≤13/2b + 1, where the lower bound holds for b≥10. We find upper and lower bounds for the more general function fc(b) which, for c = 1, improve the aforementioned upper bound to ⌈9b/4⌉ + 9, and give the same lower bound of 2b + 10. Results about fc(b) are used to find analogous results on a slightly different generalization of f⊥(b).

KW - Monochromatic sequences

KW - Van der Waerden numbers

UR - http://www.scopus.com/inward/record.url?scp=0347025898&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347025898&partnerID=8YFLogxK

U2 - 10.1016/S0012-365X(99)00137-5

DO - 10.1016/S0012-365X(99)00137-5

M3 - Article

VL - 207

SP - 137

EP - 147

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -