  We show that there is an absolute constant $c>0$ such that the following
holds. For every $n > 1$, there is a 5-uniform hypergraph on at least
$2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every
set of 6 vertices induces at most 3 edges. The double exponential growth rate
for the number of vertices is sharp. By applying a stepping-up lemma
established by the first two authors, analogous sharp results are proved for
$k$-uniform hypergraphs. This answers the penultimate open case of a conjecture
in Ramsey theory posed by Erd\H{o}s and Hajnal in 1972.
