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摘要
Let a and b be positive integers with a <= b. An (a, b)-triple is a set {x, ax + d, bx + 2d}, where x, d >= 1. Define T (a, b; r) to be the least positive integer n such that any r-coloring of {1, 2, ... , n} contains a monochromatic (a, b)-triple. Earlier results gave an upper bound on T(a, b; 2) that is a fourth degree polynomial in b and a, and a quadratic lower bound. A new upper bound for T(a, b; 2) is given that is a quadratic. Additionally, lower bounds are given for the case in which a = b, updated tables are provided, and open questions are presented.
- 出版日期2015-7
