If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k arithmetic subsequence has only terms which are $\ge$ those from $S_k$?

In other words, is it true that $\forall n, {S_k}_n \le a_n$ ? If not, can someone give a counterexample?

Also, which of these sequences are known to be small sets?