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

Tom C. Brown, Bruce M. Landman, Marni Mishna

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)149-157
Number of pages9
JournalCanadian Mathematical Bulletin
Volume40
Issue number2
DOIs
StatePublished - Jan 1 1997
Externally publishedYes

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Arithmetic sequence
Integer
Quadruple
Congruent
Colouring
Divides
Modulo
Denote
Term
Theorem
Generalization
Color

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Canadian Mathematical Bulletin, Vol. 40, No. 2, 01.01.1997, p. 149-157.

Research output: Contribution to journalArticle

Brown, Tom C. ; Landman, Bruce M. ; Mishna, Marni. / Monochromatic homothetic copies of {1, 1 + s, 1 + s + t}. In: Canadian Mathematical Bulletin. 1997 ; Vol. 40, No. 2. pp. 149-157.
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