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Recently, Kupavskii~[{\it On random subgraphs of {K}neser and {S}chrijvergraphs. J. Combin. Theory Ser. A, {\rm 2016}.}] investigated the chromaticnumber of random Kneser graphs $\KG_{n,k}(\rho)$ and proved that, in manycases, the chromatic numbers of the random Kneser graph $\KG_{n,k}(\rho)$ andthe Kneser graph $\KG_{n,k}$ are almost surely closed. He also marked thestudying 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 byintroducing an almost surely lower bound for the chromatic number of them.Roughly speaking, as a special case of our result, we show that the chromaticnumbers of the random Kneser hypergraph $\KG^r_{n,k}(\rho)$ and the Kneserhypergraph $\KG^r_{n,k}$ are almost surely closed in many cases. Moreover,restricting to the Kneser and {S}chrijver graphs, we present a purelycombinatorial proof for an improvement of Kupavskii's results.  Also, for any hypergraph $\HH$, we present a lower bound for the minimumnumber of colors required in a coloring of $\KG^r(\mathcal{H})$ with nomonochromatic $K_{t,\ldots,t}^r$ subhypergraph, where $K_{t,\ldots,t}^r$ is thecomplete $r$-uniform $r$-partite hypergraph with $t r$ vertices such that eachof its parts has $t$ vertices. This result generalizes the lower bound for thechromatic number of $\KG^r(\mathcal{H})$ found by the present authors~[{\it Onthe chromatic number of general {K}neser hypergraphs. J. Combin. Theory, Ser.B, {\rm 2015}.}].

Chromatic Number of Random Kneser Hypergraphs