On sets of type (m,n)r−1(m,n)r−1 in PG(r,q)

In recent years, a considerable effort has been directed toward the determination of parameters (k,m,n)(k,m,n) for which there exists a kk-set of type (m,n)r−1(m,n)r−1 in a projective space PG(r,q). In this paper we develop a method for determining parameters mm and nn for some fixed integer kk. As an application, we obtain a simpler proof of a well-known characterization of non-singular elliptic quadrics in PG(3,q), qq odd, and we generalize slightly two well-known characterizations: Baer subspaces in PG(4,q), qq square, and Segre varieties S1×S2S1×S2 in PG(5,q), q≥3q≥3. The method allows us to prove non-existence theorems. In particular we prove the non-existence of non-trivial qtqt-sets of type (m,n)r−1(m,n)r−1 in PG(r,q).