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, p((3n)) = 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 c < -rlnr/3 -5/18 ln r - 1/3 -r(-1/6), the random hypergraph H(n, 3, p) is strongly r-colorable with high probability and, vice versa, for c > rln r/3 - 5/18 ln r+O(ln r/r), it is not strongly r-colorable with high probability. (C) 2020 Elsevier B.V. All rights reserved.