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on Erdős #201 · additive combinatorics / arithmetic progressions

The statement

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?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.