Large gaps between consecutive prime numbers

Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erd} os, we show that G(X)> f(X) logX log logX log log log logX (log log logX) 2 ; where f(X) is a function tending to innity 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.