What is the smallest $r$ such that there is always a prime between $n$ and $rn$ for large enough $n$?

From Bertrand's postulate we know that when $n\ge2$ there is always a prime between $n$ and $2n$. From the answer here, we know that for $n\ge31$ there is always a prime between $n$ and $\frac65n$. Let $r>1$. I fully expect the following conjecture to be unproven:

There is always an $n_0$ such that when $n \ge n_0$ there is always a prime between $n$ and $rn$.

What is the smallest known $r$ for which this is known to be true?

• Why did you expect the conjecture to be unproven? – Erick Wong Feb 23 '17 at 9:37
• @ErickWong Because the $\dfrac 65n$ claim was only proven in the 1950's, so I guess I wasn't so optimistic about the general case having been proven in the past 60 years – Ovi Feb 23 '17 at 16:35
• Ah I see. So the general case was known in 1896, but the best upper bounds we have for $n_0$ as a function of $r$ are probably much weaker than the truth. There are a few plausible reasons why the first publication of $6/5$ might be so late: 1) the result isn't of intrinsic interest unless proven by elementary methods similar to Ramanujan's proof of Bertrand's postulate, 2) the explicit value of $n_0$ was arduous to verify before early computers allowed computation of Riemann zeros. – Erick Wong Feb 23 '17 at 19:10
• @Ovi : ​ In fact, see this question I asked. ​ ​ ​ ​ – user57159 May 12 '17 at 8:55

It is known to be true for all $r > 1$, see Wikipedia, in particular this means there is no such smallest $r$.
• In particular, there is no "smallest $r$", since the set of $r$ for which the claim holds does not contain its infimum. – Erick Wong Feb 23 '17 at 9:36