For a pair of integers 0 < t < r, the t-chromatic number of an
r-uniform hypergraph H=(V,E) is the minimal k, for which there exists a
partition of V into color classes U1,...,Uk such that every edge of H
intersects every color class in at most t vertices. In this paper we
determine the asymptotic behavior of the t-chromatic number of the random
r-uniform hypergraph on n vertices with edge probability p=p(n) for all
possible values of t and for a wide range of values of p.
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