Erdős and Prachar's question: do the increases of $p_n/n$ have positive lower density?

Theorem 1.1: for every fixed $C>0$ there are $c(C)>0$ and $N_0(C)$ with $\#\{n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N$ for all $N\ge N_0$, with ordinary lower density and equal weight on prime indices. Corollary 1.2 (take $C=2$) answers Erdős-Prachar: the increases of $p_n/n$ have positive lower density. The main theorem is stronger than the posed question. The constant $c(C)$ is not made explicit or compared with the Poisson prediction $e^{-C}$, and the further questions recorded with Problem #968 on $u_n<u_{n+1}<u_{n+2}$ and $u_n>u_{n+1}>u_{n+2}$ are not addressed.

Theorem 1.1: for every fixed $C>0$ there are $c(C)>0$ and $N_0(C)$ with $\#\{n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N$ for all $N\ge N_0$, with ordinary lower density and equal weight on prime indices. Corollary 1.2 (take $C=2$) answers Erdős-Prachar: the increases of $p_n/n$ have positive lower density. The main theorem is stronger than the posed question. The constant $c(C)$ is not made explicit or compared with the Poisson prediction $e^{-C}$, and the further questions recorded with Problem #968 on $u_n<u_{n+1}<u_{n+2}$ and $u_n>u_{n+1}>u_{n+2}$ are not addressed.