On the Chromatic Number of the Square of the Kneser Graph K(2k+1, k)

Abstract.The Kneser graphK(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n−2k+2. Zoltán Füredi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that χ(K2(2k+1, k))≤4k when k is odd and χ(K2(2k+1, k))≤4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on χ(K2(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.