On the strong colorings of a random 4-uniform hypergraph

The paper deals with the problem of estimating the probability threshold for theproperty of the strong colorability for a random 4-uniform hypergraph in the binomial model H(n, 4, p). A coloring of the hypergraph vertex set is said to be strong if any two vertices u ̸= v that are contained in the same edge, are given different colors. We estimate the probability threshold of the existence of a strong coloring with r colors for H(n, 4, p). The threshold corresponds to the so called sparse case when p = cn/(︀n4)︀ for fixed c > 0. We prove that for c r lnr/6 - 13/36 lnr - 1/6 - r-1/9 the random hypergraph H(n, 4, cn/(︀n4)︀) is strongly r-colorable with r colors with probability tending to 1 as n → ∞.