Arrangements of [formula omitted]-sets with intersection constraints

Abstract

Given positive integers n,k where n≥k, let f(n,k) denote the largest integer s such that there exists a cyclic ordering of the k-sets on [n]={0,1,…,n−1} such that every s consecutive k-sets are pairwise intersecting. Equivalently, f(n,k) is the largest s such that the complement K(n,k)¯ of the Kneser graph K(n,k) contains the sth power of a Hamiltonian cycle.For each n≥6 we show that f(n,2)=3. We show that f(n,3) equals either 2n−8 or 2n−7 when n is sufficiently large, conjecturing that 2n−8 is the correct value. For each k≥4 and n sufficiently large we show that 2nk−2(k−2)!−(72k−2)nk−3(k−3)!−O(nk−4)≤f(n,k)≤2nk−2(k−2)!−(72k−3.2)nk−3(k−3)!+o(nk−3).