  Essential covers were introduced by Linial and Radhakrishnan as a model that
captures two complementary properties: (1) all variables must be included and
(2) no element is redundant. In their seminal paper, they proved that every
essential cover of the $n$-dimensional hypercube must be of size at least
$\Omega(n^{0.5})$. Later on, this notion found several applications in
complexity theory. We improve the lower bound to $\Omega(n^{0.52})$, and
describe two applications.
