Problem 201

Erdős Problem #201: declared status 'open'. Formalized: no. Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\{1,\ldots,N\}$ without a $k$-term arithmetic progression? Is it true that$$\lim_{N\to \infty}\frac{R_3(N)}{G_3(N)}=1?$$ Prize: no. OEIS: A003002, A003003, A003004, A003005. Tags: additive combinatorics, arithmetic progressions.