Long gaps between primes

Let $p_n$ denotes the $n$-th prime. We prove that $$\underset{p_{n+1}\le X}{\max }(p_{n+1}-p_n)\gg \frac{\log X\log \log X\log \log \log \log X}{\log \log \log X}$$ 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.