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 Erdos, we show that G(X) ≥ f(X) log X log log X log log log log X/(log log log X)², where f(X) is a function tending to in finity 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 exis- tence and distribution of long arithmetic progressions consisting entirely of primes.