### Abstract

Numbers similar to those of van der Waerden are studied. We consider increasing sequences of positive integers {x_{1}, x_{2},..., x_{n}} that either form an arithmetic progression or for which there exists a polynomial f with integer coefficients and degree exactly n - 2, and x_{j+1} =f(x_{j}). We denote by q(n, k) the least positive integer such that if {1, 2,..., q(n, k)} is partitioned into k classes, then some class must contain a sequence of the type just described. Upper bounds are obtained for q(n, 3), q(n, 4), q(3, k), and q(4, k). A table of several values is also given.

Original language | English (US) |
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Pages (from-to) | 287-291 |

Number of pages | 5 |

Journal | Graphs and Combinatorics |

Volume | 6 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1990 |

Externally published | Yes |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Landman, B. M., & Greenwell, R. N. (1990). Some new bounds and values for van der Waerden-like numbers.

*Graphs and Combinatorics*,*6*(3), 287-291. https://doi.org/10.1007/BF01787579