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Claim. Write $G(X)$ for the largest gap between consecutive primes below $X$ and $\log_k$ for the $k$-fold iterated logarithm. R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), no. 4, 242--247, proves

$$G(X)\ge(c+o(1))\,\frac{\log X\log_2X\log_4X}{(\log_3X)^2}$$

with $c=1/3$. The value of the constant is as the introductions of Ford, Green, Konyagin and Tao (arXiv:1408.4505, p. 2) and of the 2018 paper of Ford, Green, Konyagin, Maynard and Tao (see its library card) record it. Since $\log p_n\sim\log n$, the bound gives, for every $C$ with $0<C<1/3$, infinitely many $n$ with $p_{n+1}-p_n>C\log n\log_2n\log_4n/(\log_3n)^2$, the question of Problem 4 answered yes for those $C$. Later work raised $c$ to $\tfrac12e^\gamma$ (Schönhage), $e^\gamma$ (Rankin), $1.31256e^\gamma$ (Maier and Pomerance) and $2e^\gamma$ (Pintz), before the 2014 theorems of Ford, Green, Konyagin and Tao and of Maynard made it arbitrary.

Covers. Every $C$ with $0<C<1/3$: for each such $C$ there are infinitely many $n$ with the question's inequality.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: J. London Math. Soc. 13 (1938), no. 4, 242--247, doi:10.1112/jlms/s1-13.4.242. Not reviewed: the site's commentary mentions Rankin's result, but the site's PROVED label credits the solution to Maynard and to Ford, Green, Konyagin and Tao.