MONOCHROMATIC HOMOTHETIC COPIES OF f 1 Ò 1 + s Ò 1 + s + t g TOM

For positive integers s and t, let f (sÒ t) denote the smallest positive integer N such that every 2-colouring of [1ÒN] = f1Ò 2Ò    ÒNg has a monochromatic homothetic copy of f1Ò 1 + sÒ 1 + s + tg. We show that f (sÒ t) = 4(s + t) + 1 whenever sÛg and tÛg are not congruent to 0 (modulo 4), where g = gcd(sÒ 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 f1Ò 1 + 1Ò 1 + 1 + 1g. We also show that f (sÒ t) = 4(s + t) + 1 in many other cases (for example, whenever s Ù 2t Ù 2 and t does not divide s), and that f (sÒ t) 4(s + t) + 1 for all s, t. Thus the set of homothetic copies of f1Ò 1 + sÒ 1 + s + tg 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.