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 ) . Erdős et al. asked whether the counting function, S ( A , x ) , of a Stanley sequence S ( A ) satisfies S ( A , x ) > x 1 2 − ϵ for every ϵ > 0 and x > x 0 ( ϵ , A ) . In this paper we answer this question in the affirmative; in fact, we prove the slightly stronger result that S ( A , x ) ≥ ( 2 − ϵ ) x for x ≥ x 0 ( ϵ , A ) .