On sets of type in

In recent years, a considerable effort has been directed toward the determination of parameters ( k , m , n ) for which there exists a k -set of type ( m , n ) r − 1 in a projective space PG ( r , q ) . In this paper we develop a method for determining parameters m and n for some fixed integer k . As an application, we obtain a simpler proof of a well-known characterization of non-singular elliptic quadrics in PG ( 3 , q ) , q odd, and we generalize slightly two well-known characterizations: Baer subspaces in PG ( 4 , q ) , q square, and Segre varieties S 1 × S 2 in PG ( 5 , q ) , q ≥ 3 . The method allows us to prove non-existence theorems. In particular we prove the non-existence of non-trivial q t -sets of type ( m , n ) r − 1 in PG ( r , q ) .