### Abstract

For positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1,2,....N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}. We show that f(s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden's theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + 1}. We also show that f(s, t) = 4(s + t) + 1 in many other cases (for example, whenever s > 2t > 2 and t does not divide s), and that f(s, t) ≤ 4(s + t) + 1 for all s, t. Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other "natural" sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.

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

Pages (from-to) | 149-157 |

Number of pages | 9 |

Journal | Canadian Mathematical Bulletin |

Volume | 40 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1997 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Canadian Mathematical Bulletin*,

*40*(2), 149-157. https://doi.org/10.4153/CMB-1997-018-3

**Monochromatic homothetic copies of {1, 1 + s, 1 + s + t}.** / Brown, Tom C.; Landman, Bruce M.; Mishna, Marni.

Research output: Contribution to journal › Article

*Canadian Mathematical Bulletin*, vol. 40, no. 2, pp. 149-157. https://doi.org/10.4153/CMB-1997-018-3

}

TY - JOUR

T1 - Monochromatic homothetic copies of {1, 1 + s, 1 + s + t}

AU - Brown, Tom C.

AU - Landman, Bruce M.

AU - Mishna, Marni

PY - 1997/6

Y1 - 1997/6

N2 - For positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1,2,....N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}. We show that f(s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden's theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + 1}. We also show that f(s, t) = 4(s + t) + 1 in many other cases (for example, whenever s > 2t > 2 and t does not divide s), and that f(s, t) ≤ 4(s + t) + 1 for all s, t. Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other "natural" sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.

AB - For positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1,2,....N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}. We show that f(s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden's theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + 1}. We also show that f(s, t) = 4(s + t) + 1 in many other cases (for example, whenever s > 2t > 2 and t does not divide s), and that f(s, t) ≤ 4(s + t) + 1 for all s, t. Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other "natural" sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.

UR - http://www.scopus.com/inward/record.url?scp=0031156339&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031156339&partnerID=8YFLogxK

U2 - 10.4153/CMB-1997-018-3

DO - 10.4153/CMB-1997-018-3

M3 - Article

AN - SCOPUS:0031156339

VL - 40

SP - 149

EP - 157

JO - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

IS - 2

ER -