On Arcs in a Finite Projective Plane

The aim of this paper is to generalize and unify results of B. Qvist, B. Segre, M. Sce, and others concerning arcs in a finite projective plane. The method consists of applying completely elementary combinatorial arguments. To the usual axioms for a projective plane we add the condition that the number of points be finite. Thus there exists an integer n ⩾ 2, called the order of the plane, such that the number of points and the number of lines equal n2 + n + 1 and the number of points on a line and the number of lines through a point equal n + 1. In the following, n will always denote the order of a finite plane. Desarguesian planes of order n, formed by the analytic geometry with coefficients from the Galois field of order n, are examples of finite projective planes. We shall not assume that our planes are Desarguesian, however.