### 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) |
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Pages (from-to) | 279-290 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 256 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Sep 28 2002 |

Externally published | Yes |

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### 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