On Two Limit Values of the Chromatic Number of a Random Hypergraph

Abstract

The limit concentration of the values of the chromatic number of the random hypergraph $H(n,k,p)$ in the binomial model is studied. It is proved that, for a fixed $k\ge 3$ and with not too rapidly increasing $n^{k-1}p$, the chromatic number of the hypergraph $H(n,k,p)$ lies, with probability tending to 1, in the set of two consecutive values. Moreover, it is shown that, under slightly stronger constraints on the growth of $n^{k-1}p$, these values can be explicitly evaluated as functions of $n$ and $p$.