  The independence number of a hypergraph H is the size of a largest set of
vertices containing no edge of H. In this paper, we prove new sharp bounds on
the independence number of n-vertex (r+1)-uniform hypergraphs in which every
r-element set is contained in at most d edges, where 0 < d < n/(log n)^{3r^2}.
Our relatively short proof extends a method due to Shearer. We give an
application to hypergraph Ramsey numbers involving independent neighborhoods.
