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 $\varepsilon>0$.

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.