### Abstract

Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers {x_{1}, x_{2}, ..., x_{n}} either that form an arithmetic sequence or for which there exists a polynomial f(x) = Σ_{i = 0}^{n - 2} a_{i}x^{i} with a_{i} ε{lunate} Z, a_{n - 2} > 0, and x_{j + 1} = f(x_{j}). We denote by q(n) the least positive integer such that if {1, 2, ..., q(n)} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q(n), as well as values of q(n) for n ≤ 5. A stronger upper bound for q(n) is conjectured and is shown to imply the existence of a similar bound on the nth van der Waerden number.

Original language | English (US) |
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Pages (from-to) | 82-86 |

Number of pages | 5 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 50 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1989 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

**On the existence of a reasonable upper bound for the van der Waerden numbers.** / Greenwell, Raymond N.; Landman, Bruce M.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 50, no. 1, pp. 82-86. https://doi.org/10.1016/0097-3165(89)90006-X

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TY - JOUR

T1 - On the existence of a reasonable upper bound for the van der Waerden numbers

AU - Greenwell, Raymond N.

AU - Landman, Bruce M.

PY - 1989/1/1

Y1 - 1989/1/1

N2 - Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers {x1, x2, ..., xn} either that form an arithmetic sequence or for which there exists a polynomial f(x) = Σi = 0n - 2 aixi with ai ε{lunate} Z, an - 2 > 0, and xj + 1 = f(xj). We denote by q(n) the least positive integer such that if {1, 2, ..., q(n)} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q(n), as well as values of q(n) for n ≤ 5. A stronger upper bound for q(n) is conjectured and is shown to imply the existence of a similar bound on the nth van der Waerden number.

AB - Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers {x1, x2, ..., xn} either that form an arithmetic sequence or for which there exists a polynomial f(x) = Σi = 0n - 2 aixi with ai ε{lunate} Z, an - 2 > 0, and xj + 1 = f(xj). We denote by q(n) the least positive integer such that if {1, 2, ..., q(n)} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q(n), as well as values of q(n) for n ≤ 5. A stronger upper bound for q(n) is conjectured and is shown to imply the existence of a similar bound on the nth van der Waerden number.

UR - http://www.scopus.com/inward/record.url?scp=38249023360&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249023360&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(89)90006-X

DO - 10.1016/0097-3165(89)90006-X

M3 - Article

AN - SCOPUS:38249023360

VL - 50

SP - 82

EP - 86

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -