Long gaps between primes

Abstract:

Let $p_n$ denote the $n$th prime. We prove that \[ \max _{p_{n} \leqslant X} (p_{n+1}-p_n) \gg \frac {\log X \log \log X\log \log \log \log X}{\log \log \log X}\] for sufficiently large $X$, improving upon recent bounds of the first, second, third, and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.