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.




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**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.*



3-term arithmetic progressions of terms as frequent as primes
It is open. Best current results in the quantitative version of Roth's theorem belong to Sanders and allow to find 3-term arithmetic progression between something like $O(n/\log^{1-\varepsilon} n)$ numbers not exceeding $n$, for any given $\varepsilon>0$.


UPD: already not to Sanders, but to Bloom (see quid's comment), however the estimates are still weaker than for primes. 


