Let's call a sequence $k^+$-free if it is contains no arithmetic subsequence of length $k$. Define the $\bf{density}$ of a sequence of natural numbers $s_n$ as $$\lim_{n\to \infty} \frac{n}{s_n}$$
Do all $k^+$-free sequences have density $0$?
All geometric series $s_n = i^n$ for $i \ge 2$ are $3$-sequences, for example. It seems that these are the "densest" possible $3$-sequences. This observation is consistent with the above.
Perhaps it is possible to construct a proof using strong induction.
EDIT: I'm reviving this question because I actually would like to see the solution for the case when we require that there be no arithmetic subsequence of length $k$.