On the growth of the counting function of Stanley sequences

Given a finite set of nonnegative integers A with no three-term arithmetic progressions, the Stanley sequence generated by A, denoted as S(A), is the infinite set created by beginning with A and then greedily including strictly larger integers which do not introduce a three-term arithmetic progression in S(A). Erd?s et al. asked whether the counting function, S(A,x), of a Stanley sequence S(A) satisfies for every ?>0 and x>x₀(?,A). In this paper we answer this question in the affirmative; in fact, we prove the slightly stronger result that for x≥x₀(?,A).