Chromatic Number of Random Kneser Hypergraphs

Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijver graphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromatic number of random Kneser graphs $\KG_{n,k}(\rho)$ and proved that, in many cases, the chromatic numbers of the random Kneser graph $\KG_{n,k}(\rho)$ and the Kneser graph $\KG_{n,k}$ are almost surely closed. He also marked the studying of the chromatic number of random Kneser hypergraphs $\KG^r_{n,k}(\rho)$ as a very interesting problem. With the help of $\Z_p$-Tucker lemma, a combinatorial generalization of the Borsuk-Ulam theorem, we generalize Kupavskii's result to random general Kneser hypergraphs by introducing an almost surely lower bound for the chromatic number of them. Roughly speaking, as a special case of our result, we show that the chromatic numbers of the random Kneser hypergraph $\KG^r_{n,k}(\rho)$ and the Kneser hypergraph $\KG^r_{n,k}$ are almost surely closed in many cases. Moreover, restricting to the Kneser and {S}chrijver graphs, we present a purely combinatorial proof for an improvement of Kupavskii's results. Also, for any hypergraph $\HH$, we present a lower bound for the minimum number of colors required in a coloring of $\KG^r(\mathcal{H})$ with no monochromatic $K_{t,\ldots,t}^r$ subhypergraph, where $K_{t,\ldots,t}^r$ is the complete $r$-uniform $r$-partite hypergraph with $t r$ vertices such that each of its parts has $t$ vertices. This result generalizes the lower bound for the chromatic number of $\KG^r(\mathcal{H})$ found by the present authors~[{\it On the chromatic number of general {K}neser hypergraphs. J. Combin. Theory, Ser. B, {\rm 2015}.}].