Panchromatic colorings of random hypergraphs (Q1996843)

This paper studies panchromatic colorings of random hypergraphs. Let \(H(n,k,p)\) be the binomial model of a random \(k\)-homogeneous hypergraph over \(n\) vertices with edge probability \(p\). It is shown that there exists a natural number \(k_0\) such that for each \(k>k_0\), \(3\le r<0.1\sqrt{k}\) and \(c<[(\ln r)/r][r/(r-1)]^k-[(\ln r)/2]-20k^2\ln r\big\{[r/(r-1)]\max(r^{-1/(r-1)},(r/(r+1))^2)\big\}^k\) the random hypergraph \(H(n,k,cn/\binom{n}{k})\) may be panchromatically colored with \(r\) colors almost surely as \(n\) tends to infinity. There also exists a constant \(C>0\) such that if under the same conditions \(c>[(\ln r)/r][r/(r-1)]^k-[(\ln r)/2]+C\ln r\{r(r-2)/(r-1)^2\}^k\), there does not exist a panchromatic \(r\)-coloring of the graph \(H(n,k,cn/\binom{n}{k})\) almost surely.