On small blocking sets and their linearity

Abstract

We prove that a small minimal blocking set of PG ( 2 , q ) is “very close” to be a linear blocking set over some subfield GF ( p e ) < GF ( q ) . This implies that (i) a similar result holds in PG ( n , q ) for small minimal blocking sets with respect to k-dimensional subspaces ( 0 ⩽ k ⩽ n ) and (ii) most of the intervals in the interval-theorems of Szőnyi and Szőnyi–Weiner are empty.