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.