For a given positive integer k, sr(m, k) denotes the minimal positive integer such that every coloring of [n], n ≥ sr(m, k), that uses each color at most k times, yields a rainbow AP (m); that is, an m-term arithmetic progression, all of whose terms receive different colors. We prove that 17 8 k +O(1) ≤ sr(3, k) ≤ 15 7 k + O(1) and sr(m, 2) > ⌊ 2 2 ⌋, improving the previous bounds of Alon, Caro...

This paper studies the nonlinear optimization problems subject to bipolar max-min fuzzy relation equation constraints. The feasible solution set of the problems is non-convex, in a general case. Therefore, conventional nonlinear optimization methods cannot be ideal for resolution of such problems. Hence, a Genetic Algorithm (GA) is proposed to find their optimal solution. This algorithm uses th...