Least prime $p_n$ such that $p_{n+1}-p_n\geq k$

Question

I define $\rho(k)$ to be the least prime $p_n$ such that $p_{n+1}-p_n\geq k$.

Since every number in the set $\{(k+1)!+2, (k+1)!+3, (k+1)!+4, \cdots, (k+1)!+(k+1)\}$ is composite, we have that $\rho(k)\leq (k+1)!+1$.

Are there better bounds for it?

Thanks.

Answer 1

It would appear that there are not going to be results that apply for small numbers; everything is about "sufficiently large" numbers. However, see

KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, JAMES MAYNARD, AND TERENCE TA0

https://arxiv.org/pdf/1412.5029v3.pdf