Large gaps between primes

We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log x)(\log \log x)(\log \log \log \log x)(\log \log \log x{)}^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.