Arrangements of -sets with intersection constraints

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 s th power of a Hamiltonian cycle. ch n ≥ 6 we show that f ( n , 2 ) = 3 . We show that f ( n , 3 ) equals either 2 n − 8 or 2 n − 7 when n is sufficiently large, conjecturing that 2 n − 8 is the correct value. For each k ≥ 4 and n sufficiently large we show that 2 n k − 2 ( k − 2 ) ! − ( 7 2 k − 2 ) n k − 3 ( k − 3 ) ! − O ( n k − 4 ) ≤ f ( n , k ) ≤ 2 n k − 2 ( k − 2 ) ! − ( 7 2 k − 3.2 ) n k − 3 ( k − 3 ) ! + o ( n k − 3 ) .