  Fix $k \geq 3$, and let $G$ be a $k$-uniform hypergraph with maximum degree
$\Delta$. Suppose that for each $l = 2, ..., k-1$, every set of l vertices of G
is in at most $\Delta^{(k-l)/(k-1)}/f$ edges. Then the chromatic number of $G$
is $O( (\Delta/\log f)^{1/(k-1)})$. This extends results of Frieze and the
second author and Bennett and Bohman. A similar result is proved for 3-uniform
hypergraphs where every vertex lies in few triangles. This generalizes a result
of Alon, Krivelevich, and Sudakov, who proved the result for graphs.
  Our main new technical contribution is a deviation inequality for positive
random variables with expectation less than 1. This may be of independent
interest and have further applications.
