Bertrand's Postulate asserts that there is a prime between $n$ and $2n$.
Is this the best such upper bound on prime gaps known today, or have stronger estimates been proved? I mean results of the kind:
- there will always be a prime between $n$ and $2n-2$, or
- there will always be a prime between $n$ and $cn$ with $1<c<2$?
Was any such improvement proved rigorously, or is Bertrand's Postulate still the best we have?
Yes. See theorem 3 of:
O. Ramar´e, Y. Saouter, Short effective intervals containing primes, J. Number Theory 98 (2003), no. 1.
A related statement states for all $\epsilon > 0$, for sufficiently large $n$, there is a prime between $n, (1+ \epsilon)n$ which is a trivial corollary of the PNT.
For your first question, both of those statements have been proven.