### Abstract

Functions analogous to the van der Waerden numbers w(n, k) are considered. We replace the class of arithmetic progressions, A, by a class A′, with A ⊂A′; thus, the associated van der Waerden-like number will be smaller for si’. We consider increasing sequences of positive integers x_{1},…, x_{n} which are either arithmetic progressions or for which there exists a polynomial φ(x) with integer coefficients satisfying φ(xi) = x_{i+1}, i = 1,…, n - 1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w′(n, k) for the general pair (n, k). A table of several new computer-generated values of these functions is provided.

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

Pages (from-to) | 177-184 |

Number of pages | 8 |

Journal | Graphs and Combinatorics |

Volume | 9 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

**An upper bound for van der Waerden-like numbers using k colors.** / Landman, Bruce M.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 9, no. 2, pp. 177-184. https://doi.org/10.1007/BF02988304

}

TY - JOUR

T1 - An upper bound for van der Waerden-like numbers using k colors

AU - Landman, Bruce M.

PY - 1993/1/1

Y1 - 1993/1/1

N2 - Functions analogous to the van der Waerden numbers w(n, k) are considered. We replace the class of arithmetic progressions, A, by a class A′, with A ⊂A′; thus, the associated van der Waerden-like number will be smaller for si’. We consider increasing sequences of positive integers x1,…, xn which are either arithmetic progressions or for which there exists a polynomial φ(x) with integer coefficients satisfying φ(xi) = xi+1, i = 1,…, n - 1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w′(n, k) for the general pair (n, k). A table of several new computer-generated values of these functions is provided.

AB - Functions analogous to the van der Waerden numbers w(n, k) are considered. We replace the class of arithmetic progressions, A, by a class A′, with A ⊂A′; thus, the associated van der Waerden-like number will be smaller for si’. We consider increasing sequences of positive integers x1,…, xn which are either arithmetic progressions or for which there exists a polynomial φ(x) with integer coefficients satisfying φ(xi) = xi+1, i = 1,…, n - 1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w′(n, k) for the general pair (n, k). A table of several new computer-generated values of these functions is provided.

UR - http://www.scopus.com/inward/record.url?scp=0010747039&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010747039&partnerID=8YFLogxK

U2 - 10.1007/BF02988304

DO - 10.1007/BF02988304

M3 - Article

VL - 9

SP - 177

EP - 184

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 2

ER -