  Let $p_n$ denotes the $n$-th prime. We prove that $$\max_{p_{n+1} \leq X}
(p_{n+1}-p_n) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log
X}$$ for sufficiently large $X$, improving upon recent bounds of the first
three 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 R\"odl nibble method.
