Let $\ \Pi(x)\ $ be the number of primes $\ p \le x$.

Let $\ x_1 < x_2 < \ldots\ $ be an arbitrary increasing sequence of positive integers such that $\ n \ge \Pi(x_{n+1}-1)\ $ 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?

My problem is related to a famous Klaus Roth's theorem.

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.

Let $\ \Pi(x)\ $ be the number of primes $\ p \le x$.

Let $\ x_1 < x_2 < \ldots\ $ be an arbitrary increasing sequence of positive integers such that $\ n \ge \Pi(x_{n+1}-1)\ $ for every $\ n=1\,\ 2\ \ldots$.

QUESTION: Does sequence $\ (x_1 < x_2 < \ldots)\ $ contain a 3-term arithmetic progression of its three consecutive members? Does there exist an infinite number of such 3-term arithmetic progressions?

My problem is related to a famous Klaus Roth's theorem.

Let $\ \Pi(x)\ $ be the number of primes $\ p \le x$.

Let $\ x_1 < x_2 < \ldots\ $ be an arbitrary increasing sequence of positive integers such that $\ n \ge \Pi(x_{n+1}-1)\ $ 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?

My problem is related to a famous Klaus Roth's theorem.