Sharp bounds for the chromatic number of random Kneser graphs

Given positive integers n⩾2k, the Kneser graph KGn,k is a whose vertex set collection of all k-element subsets {1,…,n}, with edges connecting pairs disjoint sets. One classical results in combinatorics, conjectured by and proved Lovász, states that chromatic number equal to n−2k+2. In this paper, we study random KGn,k(p), is, obtained from including each independently probability p. We prove that, for any fixed k⩾3, χ(KGn,k(1/2))=n−Θ(log2⁡n2k−2), as well χ(KGn,2(1/2))=n−Θ(log2⁡n⋅log2⁡log2⁡n2). also k⩾(1+ε)log⁡log⁡n, have χ(KGn,k(1/2))⩾n−2k−10. This significantly improves previous on subject, Kupavskii Alishahi Hajiabolhassan. The bound k second result tight up constant. discuss an interesting connection extremal problem embeddability complexes.