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Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}

Published online by Cambridge University Press:  20 November 2018

Tom C. Brown ,
Bruce M. Landman  and
Marni Mishna
Tom C. Brown
Affiliation:
Department of Mathematics and Statistics Simon Fraser University Burnaby, BC V5A 1S6, tbrown@sfu.ca
Bruce M. Landman
Affiliation:
Department of Mathematical Sciences University of North Carolina Greensboro, NC 27412, landman@steffi.uncg.edu
Marni Mishna
Affiliation:
Faculty of Mathematics University of Waterloo Waterloo, ON N2L 3G1, mjmishna@undergrad.math.uwaterloo.ca
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Abstract

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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 + t}. 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.

Keywords

05D10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 2 , 01 June 1997 , pp. 149 - 157
Copyright
Copyright © Canadian Mathematical Society 1997

References

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