  We study the problem of constructing a (near) uniform random proper
$q$-coloring of a simple $k$-uniform hypergraph with $n$ vertices and maximum
degree $\Delta$. (Proper in that no edge is mono-colored and simple in that two
edges have maximum intersection of size one). We show that if $q\geq
\max\{C_k\log n,500k^3\Delta^{1/(k-1)}\}$ then the Glauber Dynamics will become
close to uniform in $O(n\log n)$ time, given a random (improper) start. This
improves on the results in Frieze and Melsted [5].
