On non-averaging sets of integers

Let 5 be a set of positive integers no three of which are in arithmetical progression, i.e., if A, B, C are distinct elements of S, A + B ≠ 2C. We call such a set a non-averaging set. Let v(n) denote the maximum number of elements not exceeding n in any non-averaging set. The problem of finding bounds for v(n) has been treated by several authors [1, 3, 5, 6, 7]. The question first arose in connection with a theorem of van der Waerden [8].