Given positive integers $n⩾2k$, the Kneser graph $K{G}_{n,k}$ is a graph whose vertex set is the collection of all k-element subsets of the set $\{1,\dots ,n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lovász, states that the chromatic number of $K{G}_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of the random Kneser graph $K{G}_{n,k}(p)$, that is, the graph obtained from $K{G}_{n,k}$ by including each of the edges of $K{G}_{n,k}$ independently and with probability p.

We prove that, for any fixed $k⩾3$, $\chi (K{G}_{n,k}(1/2))=n-\mathrm{\Theta }(\sqrt[2k-2]{{\log }_{2}n})$, as well as $\chi (K{G}_{n,2}(1/2))=n-\mathrm{\Theta }(\sqrt[2]{{\log }_{2}n\cdot {\log }_2{\log }_{2}n})$. We also prove that, for $k⩾(1+\epsilon )\log \log n$, we have $\chi (K{G}_{n,k}(1/2))⩾n-2k-10$. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. The bound on k in the second result is also tight up to a constant. We also discuss an interesting connection to an extremal problem on embeddability of complexes.

给定正整数$n⩾2ķ$, Kneser 图 $ķG_{n,ķ}$是一个图，其顶点集是该集合的所有k元素子集的集合$\{1,\dots ,n\}$，边连接成对的不相交集。由 Kneser 推测并由 Lovász 证明的组合数学的经典结果之一指出，$ķG_{n,ķ}$等于$n-2ķ+2$. 在本文中，我们研究了随机 Kneser 图的色数 $ķG_{n,ķ}(p)$，即从得到的图$ķG_{n,ķ}$通过包括每个边缘$ķG_{n,ķ}$独立且具有概率p。

我们证明，对于任何固定的$ķ⩾3$,$\chi (ķG_{n,ķ}(1/2))=n-\mathrm{\theta }(\sqrt[2ķ-2]{{\mathrm{日志}}_{2}n})$， 也$\chi (ķG_{n,2}(1/2))=n-\mathrm{\theta }(\sqrt[2]{{\mathrm{日志}}_{2}n\cdot {\mathrm{日志}}_2{\mathrm{日志}}_{2}n})$. 我们还证明，对于$ķ⩾(1+\epsilon )\mathrm{日志}\mathrm{日志}n$， 我们有$\chi (ķG_{n,ķ}(1/2))⩾n-2ķ-10$. 这显着改善了先前由 Kupavskii 以及 Alishahi 和 Hajiabolhassan 获得的关于该主题的结果。第二个结果中k的界限也紧到一个常数。我们还讨论了与复合物可嵌入性的极端问题的有趣联系。