### 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 1989 |

Externally published | Yes |

### ASJC Scopus subject areas

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