Blocking Sets in Desarguesian Affine and Projective Planes

Abstract

In this paper we show that blocking sets of cardinality less than 3(q ϩ 1)/2 (q ϭ p n ) in Desarguesian projective planes intersect every line in 1 modulo p points. It is also shown that the cardinality of a blocking set must lie in a few relatively short intervals. This is similar to previous results of Ré dei, which were proved for a special class of blocking sets. In the particular case q ϭ p 2 , the above result implies that a nontrivial blocking set either contains a Baer-subplane or has size at least 3(q ϩ 1)/2; and this result is sharp. As a by-product, new proofs are given for the Jamison, Brouwer-Schrijver theorem on blocking sets in Desarguesian affine planes, and for Blokhuis' theorem on blocking sets in Desarguesian projective planes.

On multiple blocking sets in Galois planes

“…This latter t (mod p) result was already proven by Szőnyi [13] for minimal 1-fold blocking sets in PG(2, q), q = p n , p prime, with |B| < 3(q + 1)/2.…”

“…By replacing the line at infinity by an other line through (∞), it is possible to prove that all lines not through (∞) [13] or rather the improvement presented in [14]. Proposition 3.6 Assume that c + t < (q + 3)/2 and let H(U, V ) be an absolutely irreducible component of F 0 , which can be written as H(U, V ) = x(U p e , V ) with x U ≡ 0.…”

A characterization of multiple (n – k)-blocking sets in projective spaces of square order

“…Theorem 1.4 Let B be a small minimal 1-fold blocking set in PG(2, q), q = p h , p prime, h ≥ 1. Then B intersects every line in 1 (mod p) points, so for the exponent e of B, we have 1 ≤ e ≤ h. (Szőnyi [18])…”

A non‐existence result on Cameron–Liebler line classes

“…• (Szőnyi [16]) A 1-fold blocking set B in PG(2, q), of size |B| < q + q+3 2 , where q = p h , p prime, h 1, is uniquely reducible to a minimal blocking set B intersecting every line in 1 (mod p) points.…”