On regular {v, n}‐arcs in finite projective spaces

Abstract

A regular {v,n}-arc of a projective space P of order q is a set S of Y points such that each line of P has exactly 0 , l or n points in common with S and such that there exists a line of P intersecting S in exactly n points. Our main results are as follows: (1) If P is a projective plane of order q and if S is a regular {v, n}-arc with n 2 f i + 1, then S is a set of n collinear points, a Baer subplane, a unital, or a maximal arc. (2) If P is a projective space of order q and if S is a regular {v,n}-arc with n 2 f i + 1 spanning a subspace U of dimension at least 3, then S is a Baer subspace of U, an affine space of order q in U, or S equals the point Set Of u. 0 1993 John Wiley & Sons, Inc.

The Characterization of Cones as Pointsets with 3 Intersection Numbers

“…Identifying a geometric structure from its intersection numbers is a classical problem which is often difficult to solve. A lot of research has gone into characterizing sets with two intersection numbers (see [2,16,20,21,24,25]). However less is known about sets with more than two intersection numbers (see [3,12,17,18,22,26]).…”

The type of a point and a characterization of the set of external points of a conic inPG(2,q),qodd

“…In this case, such integers are the intersection numbers of K, see [2], [4], [7] and [14]. Since incidence is one of the earliest properties known to both mankind and mathematicians, a natural and important combinatorial problem is to classify sets with given intersection numbers, see, for istance, [1], [8], [9], [10], [15], [17], [21], [22], [23], [24], and [25].…”