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

Abstract

Given positive integers2 , a Kneser graph , is a graph whose vertex set is the collection of all -element subsets of the set {1, . . . , }, with edges connecting pairs of disjoint sets. One of the classical results in combinatorics, conjectured by M. Kneser and proved by L. Lovász, states that the chromatic number of , is equal to − 2 + 2. In this paper, we study the random Kneser graph , ( ), that is, the graph obtained from , by including each of the edges of , independently and with probability .We prove that, for any fixed 3, ( , (1/2)) = − Θ( 2 −2 √︀ log 2 ) and, for = 2, ( , (1/2)) = − Θ( 2 √︀ log 2 · log 2 log 2 ). We also prove that, for any fixed 6 and 2 −3 √ log , we have ( , (1/2)) − 2 + 2 − 2 , where = ( ) is an absolute constant. 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.

On the random version of the Erdős matching conjecture

“…As a fast growing branch of hypergraph theory, many articles are recently devoted to investigating the properties of random Kneser hypergraphs KG r n,k (p); see [1,2,3,4,6,8,9,20,21,22,25,26,27,28]. Extending some results in [3,4], Bollobás, Narayanan and Raigorodskii [6] studied the independence number of random Kneser graphs KG n,k (p).…”

Random Kneser Graphs and Hypergraphs

“…The following question seems to be worth exploring: are there any interesting classes of graphs or hypergraphs, for which the topological bounds (as the ones proven in [26] and [1]) work, while the present combinatorial approach fails? Note: New results on the subject, including the the resolution of a slightly weaker version of the conjecture above, are going to appear in [25].…”

The chromatic number of random Borsuk graphs

“…This contrasts with the situation studied by Kahle in [8], where topological lower bounds are not efficient for the chromatic number of Erdős-Rényi random graphs. Similar problems have also been studied for random Kneser graphs in [10,11].…”