Sharp bounds for the chromatic number of random Kneser graphs

TL;DR

This paper establishes new bounds on the chromatic number of random Kneser graphs, showing it is close to the classical value with specific logarithmic corrections, and improves previous results significantly.

Contribution

It provides sharp bounds for the chromatic number of random Kneser graphs for various parameters, advancing understanding beyond prior work.

Findings

For fixed k≥3, chromatic number is n minus a root of log n.

For k=2, chromatic number is n minus a different logarithmic term.

For large k, the chromatic number is at least n minus 2k minus 10.

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.