The Hilton–Milner theorem for attenuated spaces

Abstract Let V be an ( n + l ) -dimensional vector space over the finite field F q with l ≥ n > 0 and W be a fixed l -dimensional subspace of V . We say that an m -dimensional subspace U of V is of type ( m , k ) if dim ( U ∩ W ) = k . Denote the set of all subspaces of type ( m , k ) in V by M ( m , k ; n + l , n ) . The collection of all the subspaces of types ( m , 0 ) in V , where 0 ≤ m ≤ n , is the attenuated space. In this paper, we prove the Hilton–Milner theorem for M ( m , 0 ; n + l , n ) , where 3 ≤ m ≤ n .