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

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.