### Abstract

Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2,...,w(k,r)} must contain a monochromatic k-term arithmetic progression {x,x + d, x + 2d,...,x + (k - 1)d}. We investigate the following generalization of w(3,r). For fixed positive integers a and b with a ≤ b, define N(a, b; r) to be the least positive integer, if it exists, such that any r-coloring of {1,2,...,N(a,b;r)} must contain a monochromatic set of the form {x,ax + d,bx + 2d}. We show that N(a,b;2) exists if and only if b≠2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a,b), N(a,b;r) does not exist for r sufficiently large. We also give a result on sets of the form {x,ax + d,ax + 2d,...,ax + (k - 1)d}.

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

Pages (from-to) | 279-290 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 256 |

Issue number | 1-2 |

DOIs | |

State | Published - Sep 28 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

- Arithmetic progressions
- Van der Waerden

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*256*(1-2), 279-290. https://doi.org/10.1016/S0012-365X(01)00436-8

**On generalized van der waerden triples.** / Landman, Bruce M.; Robertson, Aaron.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 256, no. 1-2, pp. 279-290. https://doi.org/10.1016/S0012-365X(01)00436-8

}

TY - JOUR

T1 - On generalized van der waerden triples

AU - Landman, Bruce M.

AU - Robertson, Aaron

PY - 2002/9/28

Y1 - 2002/9/28

N2 - Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2,...,w(k,r)} must contain a monochromatic k-term arithmetic progression {x,x + d, x + 2d,...,x + (k - 1)d}. We investigate the following generalization of w(3,r). For fixed positive integers a and b with a ≤ b, define N(a, b; r) to be the least positive integer, if it exists, such that any r-coloring of {1,2,...,N(a,b;r)} must contain a monochromatic set of the form {x,ax + d,bx + 2d}. We show that N(a,b;2) exists if and only if b≠2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a,b), N(a,b;r) does not exist for r sufficiently large. We also give a result on sets of the form {x,ax + d,ax + 2d,...,ax + (k - 1)d}.

AB - Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of {1,2,...,w(k,r)} must contain a monochromatic k-term arithmetic progression {x,x + d, x + 2d,...,x + (k - 1)d}. We investigate the following generalization of w(3,r). For fixed positive integers a and b with a ≤ b, define N(a, b; r) to be the least positive integer, if it exists, such that any r-coloring of {1,2,...,N(a,b;r)} must contain a monochromatic set of the form {x,ax + d,bx + 2d}. We show that N(a,b;2) exists if and only if b≠2a, and provide upper and lower bounds for it. We then show that for a large class of pairs (a,b), N(a,b;r) does not exist for r sufficiently large. We also give a result on sets of the form {x,ax + d,ax + 2d,...,ax + (k - 1)d}.

KW - Arithmetic progressions

KW - Van der Waerden

UR - http://www.scopus.com/inward/record.url?scp=31244434719&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=31244434719&partnerID=8YFLogxK

U2 - 10.1016/S0012-365X(01)00436-8

DO - 10.1016/S0012-365X(01)00436-8

M3 - Article

VL - 256

SP - 279

EP - 290

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-2

ER -