Let $\ p_1\ < p_2 < \ldots\ $ be the sequence of all primes $\ (2\ 3\ 5\ \ldots)$.

Let $\ x_1 < x_2 < \ldots\ $ be an arbitrary increasing sequence of positive integers such that $\ x_n\le p_n\ $ for every $\ n=1\ 2\ldots\,$.

**QUESTION:** Does sequence $\ (x_1 < x_2 < \ldots)\ $ contain a 3-term arithmetic progression (of not necessary three consecutive members)? Does there exist an infinite number of such 3-term arithmetic progressions?

**Acknowledgement:** The simple version $\ x_n\le p_n\ $ of the assumption of this conjecture was provided by @zeb in a response to my equivalent original assumption which was clumsy and harder to read.

**Reference:** my MO-problem is related to a famous Klaus Roth's theorem.

*Now I see from the Fedor's answer that this was indeed essentially only an MO-problem, and otherwise not essentially original.*

notcontain 3 consecutive members in AP? $\endgroup$ – Nick Gill Dec 8 '15 at 8:55