Non-Fano Quads in Finite Projective Planes

Abstract

Given a finite projective plane of order \(n\). A quadrangle consists of four points \(A, B, C, D\), no three collinear. If the diagonal points are non-collinear, the quadrangle is called a non-Fano quad. A general sum of squares theorem is proved for the distribution of points in a plane containing a non-Fano quad, whenever \(n \geq 7\). The theorem implies that the number of possible distributions of points in a plane of order \(n\) is bounded for all \(n \geq 7\). This is used to give a simple combinatorial proof of the uniqueness of \(PP(7)\).