### Abstract

Ramsey functions similar to the van der Waerden numbers w(n) are studied. If A' is a class of sequences which includes the n-term arithmetic progressions, then we define w'(n) to be the least positive integer guaranteeing that if {1,2,...,w'(n)} is 2-colored, then there exists a monochromatic member of A'. We consider increasing sequences of positive integers {x_{1},...,x_{n}} which are either arithmetic progressions or for which there exists a polynomial p(x) with integer coefficients satisfying p(x_{i}) = x_{i+1}. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w'(n). In addition, it is shown that the existence of somewhat stronger bounds on w'(n) would imply similar bounds for w(n).

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

Pages (from-to) | 265-278 |

Number of pages | 14 |

Journal | Discrete Mathematics |

Volume | 102 |

Issue number | 3 |

DOIs | |

State | Published - May 22 1992 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

**Ramsey functions related to the van der waerden numbers.** / Landman, Bruce M.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 102, no. 3, pp. 265-278. https://doi.org/10.1016/0012-365X(92)90120-5

}

TY - JOUR

T1 - Ramsey functions related to the van der waerden numbers

AU - Landman, Bruce M.

PY - 1992/5/22

Y1 - 1992/5/22

N2 - Ramsey functions similar to the van der Waerden numbers w(n) are studied. If A' is a class of sequences which includes the n-term arithmetic progressions, then we define w'(n) to be the least positive integer guaranteeing that if {1,2,...,w'(n)} is 2-colored, then there exists a monochromatic member of A'. We consider increasing sequences of positive integers {x1,...,xn} which are either arithmetic progressions or for which there exists a polynomial p(x) with integer coefficients satisfying p(xi) = xi+1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w'(n). In addition, it is shown that the existence of somewhat stronger bounds on w'(n) would imply similar bounds for w(n).

AB - Ramsey functions similar to the van der Waerden numbers w(n) are studied. If A' is a class of sequences which includes the n-term arithmetic progressions, then we define w'(n) to be the least positive integer guaranteeing that if {1,2,...,w'(n)} is 2-colored, then there exists a monochromatic member of A'. We consider increasing sequences of positive integers {x1,...,xn} which are either arithmetic progressions or for which there exists a polynomial p(x) with integer coefficients satisfying p(xi) = xi+1. Various further restrictions are placed on the types of polynomials allowed. Upper bounds are given for the corresponding functions w'(n). In addition, it is shown that the existence of somewhat stronger bounds on w'(n) would imply similar bounds for w(n).

UR - http://www.scopus.com/inward/record.url?scp=38249011460&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249011460&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(92)90120-5

DO - 10.1016/0012-365X(92)90120-5

M3 - Article

AN - SCOPUS:38249011460

VL - 102

SP - 265

EP - 278

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 3

ER -