### Abstract

Ramsey numbers similar to those of van der Waerden are examined. Rather than considering arithmetic sequences, we look at increasing sequences of positive integers {x_{1},x_{2},...,x_{n}} for which there exists a polynomial {cauchy integral}(x)=∑^{ri}_{=0}a_{i}x ^{i}, with a_{i}ε{lunate}Z and x_{j+1}={cauchy integral}(x_{j}). We denote by p_{r}(n) the least positive integer such that if [1,2,...,p_{r}(n)] is 2-colored, then there exists a monochromatic sequence of length n generated by a polynomial of degree ≤r. We give values for p_{r}(n) for n≤5, as well as lower bounds for p_{1}(n) and p_{2}(n). We also give an upper bound for certain Ramsey numbers that are in between p_{n-2}(n) and the nth van der Waerden number.

Original language | English (US) |
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Pages (from-to) | 77-83 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 68 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1988 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*68*(1), 77-83. https://doi.org/10.1016/0012-365X(88)90043-X

**Values and bounds for Ramsey numbers associated with polynomial iteration.** / Landman, Bruce M.; Greenwell, Raymond N.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 68, no. 1, pp. 77-83. https://doi.org/10.1016/0012-365X(88)90043-X

}

TY - JOUR

T1 - Values and bounds for Ramsey numbers associated with polynomial iteration

AU - Landman, Bruce M.

AU - Greenwell, Raymond N.

PY - 1988/1/1

Y1 - 1988/1/1

N2 - Ramsey numbers similar to those of van der Waerden are examined. Rather than considering arithmetic sequences, we look at increasing sequences of positive integers {x1,x2,...,xn} for which there exists a polynomial {cauchy integral}(x)=∑ri=0aix i, with aiε{lunate}Z and xj+1={cauchy integral}(xj). We denote by pr(n) the least positive integer such that if [1,2,...,pr(n)] is 2-colored, then there exists a monochromatic sequence of length n generated by a polynomial of degree ≤r. We give values for pr(n) for n≤5, as well as lower bounds for p1(n) and p2(n). We also give an upper bound for certain Ramsey numbers that are in between pn-2(n) and the nth van der Waerden number.

AB - Ramsey numbers similar to those of van der Waerden are examined. Rather than considering arithmetic sequences, we look at increasing sequences of positive integers {x1,x2,...,xn} for which there exists a polynomial {cauchy integral}(x)=∑ri=0aix i, with aiε{lunate}Z and xj+1={cauchy integral}(xj). We denote by pr(n) the least positive integer such that if [1,2,...,pr(n)] is 2-colored, then there exists a monochromatic sequence of length n generated by a polynomial of degree ≤r. We give values for pr(n) for n≤5, as well as lower bounds for p1(n) and p2(n). We also give an upper bound for certain Ramsey numbers that are in between pn-2(n) and the nth van der Waerden number.

UR - http://www.scopus.com/inward/record.url?scp=0010709844&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010709844&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(88)90043-X

DO - 10.1016/0012-365X(88)90043-X

M3 - Article

AN - SCOPUS:0010709844

VL - 68

SP - 77

EP - 83

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1

ER -