Large gaps between consecutive prime numbers - Ford, K; Green, B; Konyagin, S; Tao, T (Princeton University, Department of Mathematics, 2016)
Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdős, we show that $G(X)\geqslant f(X)\frac{logX log logX log log log log X}{(log log logX)^2}$, where f(X) is a function tending to infinity with X. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.