### 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 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

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

**Some new bounds and values for van der Waerden-like numbers.** / Landman, Bruce M.; Greenwell, Raymond N.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 6, no. 3, pp. 287-291. https://doi.org/10.1007/BF01787579

}

TY - JOUR

T1 - Some new bounds and values for van der Waerden-like numbers

AU - Landman, Bruce M.

AU - Greenwell, Raymond N.

PY - 1990/9/1

Y1 - 1990/9/1

N2 - Numbers similar to those of van der Waerden are studied. We consider increasing sequences of positive integers {x1, x2,..., xn} that either form an arithmetic progression or for which there exists a polynomial f with integer coefficients and degree exactly n - 2, and xj+1 =f(xj). 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.

AB - Numbers similar to those of van der Waerden are studied. We consider increasing sequences of positive integers {x1, x2,..., xn} that either form an arithmetic progression or for which there exists a polynomial f with integer coefficients and degree exactly n - 2, and xj+1 =f(xj). 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.

UR - http://www.scopus.com/inward/record.url?scp=0010657096&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010657096&partnerID=8YFLogxK

U2 - 10.1007/BF01787579

DO - 10.1007/BF01787579

M3 - Article

AN - SCOPUS:0010657096

VL - 6

SP - 287

EP - 291

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 3

ER -