### Abstract

Let 1 {less-than or slanted equal to} a {less-than or slanted equal to} b be integers. A triple of the form ( x, a x + d, b x + 2 d ), where x, d are positive integers is called an ( a, b )-triple. The degree of regularity of the family of all ( a, b )-triples, denoted dor ( a, b ), is the maximum integer r such that every r-coloring of N admits a monochromatic ( a, b )-triple. We settle, in the affirmative, the conjecture that dor ( a, b ) < ∞ for all ( a, b ) ≠ ( 1, 1 ). We also disprove the conjecture that dor ( a, b ) ∈ { 1, 2, ∞ } for all ( a, b ).

Original language | English (US) |
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Pages (from-to) | 124-128 |

Number of pages | 5 |

Journal | Advances in Applied Mathematics |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2006 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Advances in Applied Mathematics*,

*37*(1), 124-128. https://doi.org/10.1016/j.aam.2005.08.003

**On the degree of regularity of generalized van der Waerden triples.** / Frantzikinakis, Nikos; Landman, Bruce M.; Robertson, Aaron.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics*, vol. 37, no. 1, pp. 124-128. https://doi.org/10.1016/j.aam.2005.08.003

}

TY - JOUR

T1 - On the degree of regularity of generalized van der Waerden triples

AU - Frantzikinakis, Nikos

AU - Landman, Bruce M.

AU - Robertson, Aaron

PY - 2006/7/1

Y1 - 2006/7/1

N2 - Let 1 {less-than or slanted equal to} a {less-than or slanted equal to} b be integers. A triple of the form ( x, a x + d, b x + 2 d ), where x, d are positive integers is called an ( a, b )-triple. The degree of regularity of the family of all ( a, b )-triples, denoted dor ( a, b ), is the maximum integer r such that every r-coloring of N admits a monochromatic ( a, b )-triple. We settle, in the affirmative, the conjecture that dor ( a, b ) < ∞ for all ( a, b ) ≠ ( 1, 1 ). We also disprove the conjecture that dor ( a, b ) ∈ { 1, 2, ∞ } for all ( a, b ).

AB - Let 1 {less-than or slanted equal to} a {less-than or slanted equal to} b be integers. A triple of the form ( x, a x + d, b x + 2 d ), where x, d are positive integers is called an ( a, b )-triple. The degree of regularity of the family of all ( a, b )-triples, denoted dor ( a, b ), is the maximum integer r such that every r-coloring of N admits a monochromatic ( a, b )-triple. We settle, in the affirmative, the conjecture that dor ( a, b ) < ∞ for all ( a, b ) ≠ ( 1, 1 ). We also disprove the conjecture that dor ( a, b ) ∈ { 1, 2, ∞ } for all ( a, b ).

UR - http://www.scopus.com/inward/record.url?scp=33646511994&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646511994&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2005.08.003

DO - 10.1016/j.aam.2005.08.003

M3 - Article

VL - 37

SP - 124

EP - 128

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 1

ER -