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

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

Canadian mathematical bulletin, Tome 40 (1997) no. 2, pp. 149-157

Résumé

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.

Détail

DOI : 10.4153/CMB-1997-018-3

Mots-clés : 05D10