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 \(r\).