The Odd Girth of the Generalised Kneser Graph

LetX={1, 2, . . . , n}be a set ofnelements and letX(r)be the collection of all the subsets ofXcontaining preciselyrelements. Then the generalised Kneser graphK(n, r, s)(when2r-s≤n)is the graph with vertex setX(r)and edgesABforA, B∈X(r)with ‖A∩B‖≤s.Here we show that the odd girth of the generalised Kneser graphK(n, r, s)is s)/[n − 2(r − s)]⌉+1 ed thatnis large enough compared withrands.