Erdős problem 817

Let $k \geq 3$. Define $g_k(n)$ to be the minimal $N$ such that $\{1, ..., N\}$ contains some $A$ of size $|A| = n$ such that $$\langle A\rangle = \left\{\sum_{a \in A} \epsilon_a a : \epsilon_a \in\{0, 1\}\right\}$$ contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that $$g_3(n) \gg 3^n$$