On the chromatic number of a random subgraph of the Kneser graph

S. G. Kiselev, A. M. Raigorodskii
2017 Doklady. Mathematics  
Given positive integers n ≥ 2k, a Kneser graph KG n,k is a graph whose vertex set is the collection of all k-element subsets of the set {1, . . . , n}, with edges connecting pairs of disjoint sets. A famous result due to L. Lovász states that the chromatic number of KG n,k is equal to n − 2k + 2. In this paper, we study the random Kneser graph KG n,k (p), 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 ≥ 3, χ(KG
more » ... ,k (1/2)) = n − Θ( 2k−2 log 2 n). We also provide new bounds for the case of growing k. This significantly improves previous results on the subject, obtained by Kupavskii and by Alishahi and Hajiabolhassan. We also discuss an interesting connection to an extremal problem on embeddability of complexes.
doi:10.1134/s1064562417050209 fatcat:2uc7afkbsbcexi64npoh24vfie