Problem 817

Erdős Problem #817: declared status 'open'. Formalized: yes. Let $k\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\{1,\ldots,N\}$ contains some $A$ of size $\lvert A\rvert=n$ such that$$\langle A\rangle = \left\{\sum_{a\in A}\epsilon_aa: \epsilon_a\in \{0,1\}\right\}$$contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that$$g_3(n) \gg 3^n?$$ Prize: no. Tags: additive combinatorics.