On the strong chromatic number of a random 3-uniform hypergraph

This paper deals with estimating the threshold for the strong r-colorability of a random 3-uniform hypergraph in the binomial model H(n,3,p). A vertex coloring is said to be strong for a hypergraph if every two vertices sharing a common edge are colored with distinct colors. It is known that the threshold corresponds to the sparse case, when the expected number of edges is a linear function of n, pn3=cn, and c>0 depends on r, but not on n. We establish the threshold as a bound on the parameter c up to an additive constant. In particular, by using the second moment method we prove that for large enough r and crlnr3−518lnr+O(lnr∕r), it is not strongly r-colorable with high probability.