Van der Waerden’s Theorem on Homothetic Copies of {1, 1 + s, 1 + s + t}
نویسنده
چکیده
Abstract For all positive integers s and t, Brown et. al [1] defined f(s, t) to be the smallest positive integer N such that every 2-coloring of [1, N ] has a monochromatic homothetic copy of {1, 1+ s, 1+ s+ t}. They proved that f(s, t) ≤ 4(s + t) + 1 for all s, t and that the equality holds in the case where both s/g 6≡ 0 (mod 4) and t/g 6≡ 0 (mod 4) with g = gcd(s, t) and in many other cases. Also they proved that for all positive integer m, f(4mt, t) = f(t, 4mt) = 4(4mt+ t)− t+ 1 or 4(4mt+ t) + 1. In this paper, we show that f(4mt, t) = f(t, 4mt) = 4(4mt + t) − t + 1 and that for all the other (s, t), f(s, t) = 4(s + t) + 1. keywords: van der Waerden’s theorem, arithmetic progression, homothetic copy, 2−coloring, monochromatic triple
منابع مشابه
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 ari...
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