The chromatic numbers of random hypergraphs

For a pair of integers 1 < r, the -chromatic number of an r-uniform hypergraph H = (V;E) is the minimal k, for which there exists a partition of V into subsets T1; : : : ; Tk such that je\Tij for every e 2 E. In this paper we determine the asymptotic behavior of the -chromatic number of the random r-uniform hypergraph Hr(n; p) for all possible values of and for all values of p down to p = (n r+1).