Bounds on threshold probabilities for coloring properties of random hypergraphs

We study the threshold probability for the property of existence of a special-form $r$-coloring for a random $k$-uniform hypergraph in the $H(n,k,p)$ binomial model. A parametric set of $j$-chromatic numbers of a random hypergraph is considered. A coloring of hypergraph vertices is said to be $j$-proper if every edge in it contains no more than $j$ vertices of each color. We analyze the question of finding the sharp threshold probability of existence of a $j$-proper $r$-coloring for $H(n,k,p)$. Using the second moment method, we obtain rather tight bounds for this probability provided that $k$ and $j$ are large as compared to $r$.