Sharp bounds for the chromatic number of random Kneser graphs

Abstract

Given positive integers $n\ge 2k$, the {\it Kneser graph} $KG_{n,k}$ is a graph whose vertex set is the collection of all $k$-element subsets of the set $\{1,\ldots, n\}$, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by Kneser and proved by Lov\'asz, states that the chromatic number of $KG_{n,k}$ is equal to $n-2k+2$. In this paper, we study the chromatic number of the {\it random Kneser graph} $KG_{n,k}(p)$, that is, the graph obtained from $KG_{n,k}$ by including each of the edges of $KG_{n,k}$ independently and with probability $p$. We prove that, for any fixed $k\ge 3$, $\chi(KG_{n,k}(1/2)) = n-\Theta(\sqrt[2k-2]{\log_2 n})$, as well as $\chi(KG_{n,2}(1/2)) = n-\Theta(\sqrt[2]{\log_2 n \cdot \log_2\log_2 n})$. We also prove that, for $k\ge (1+\varepsilon) \log\log n$, we have $\chi(KG_{n,k}(1/2))\ge n-2k-10$. 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.