Blocking subspaces with points and hyperplanes

Abstract

In this paper, we characterise the smallest sets $B$ consisting of points and hyperplanes in $\text{PG}(n,q)$, such that each $k$-space is incident with at least one element of $B$. If $k>\frac{n-1}{2}$, then the smallest construction consists only of points. Dually, if $k<\frac{n-1}{2}$, the smallest example consists only of hyperplanes. However, if $k=\frac{n-1}{2}$, then there exist sets containing both points and hyperplanes, which are smaller than any blocking set containing only points or only hyperplanes.

7 pages. UPDATE: After publication of this paper, we found out that in case $k=\frac{n-1}{2}$, the correct lower bound and a classification of the smallest examples was already established by Blokhuis, Brouwer, and Szőnyi [A. Blokhuis, A. E. Brouwer, T. Szőnyi. On the chromatic number of $q$-Kneser graphs. Des. Codes Crytpogr. 65:187-197, 2012]