Let k and n be positive integers, and let d(n;k) be the maximum density in f0;1;2 : : : ;kn 1g of a set containing no arithmetic progression of k terms with first term a = ∑aik and common difference d = ∑εik, where 0 ai k 1, εi = 0 or 1, and εi = 1 ) ai = 0. Setting βk = limn!∞ d(n;k), we show that limk!∞ βk is either 0 or 1. Throughout, we shall use the notation [a;b) = fa;a+ 1;a+ 2; : : : ;b ...