Parameters for sets of type \((m,n)\) in projective planes of prime power order
A set \(\mathcal S\) of \(k\) points in a projective plane of order \(q\) is of type \((m,n)\), \(m < n,\) if each line intersects \(\mathcal S\) in either \(m\) or \(n\) points. The parameters \((m,n,q)\) are called standard if \(q = (n-m)^2\). The authors show that for fixed \(m > 0\) there are only finitely many possible nonstandard parameters with \(q\) a prime power. The proof is achieved by carefully analyzing a diophantine equation, due to \textit{G. Tallini} [J. Geom. 29, 191-199 (1987; Zbl 0618.51004)], which has to be satisfied by the parameters. The authors also devise a method for determining these parameters and they list them all for \(m \leq 1000\). The results are purely arithmetical and the question whether there actually exist sets of type \((m,n)\) with these parameters remains wide open.
scientific article; zbMATH DE number 1295476
scientific article