Chromatic numbers of Kneser-type graphs

Let G(n,r,s) be a graph whose vertices are all r-element subsets of an n-element set, in which two vertices are adjacent if they intersect in exactly s elements. In this paper we study chromatic numbers of G(n,r,s) with r,s being fixed constants and n tending to infinity. Using a recent result of Keevash on existence of designs we deduce an inequality [Formula presented] for r>s with r,s fixed constants. This inequality gives sharp upper bounds for r⩽2s+1. Also we develop an elementary approach to this problem and prove that [Formula presented] without use of Keevash's results. Some bounds on the list chromatic number of G(n,r,s) are also obtained.