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arXiv · 2410.07937

Blocking Planes by Lines in $\operatorname{PG}(n,q)$

Benedek Kovács·Zoltán Lóránt Nagy·Dávid R. Szabó
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Abstract

In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces $\operatorname{PG}(n,q)$ such that every plane is incident with at least one line of the set. This is the first main open problem concerning the minimum size of $(s,t)$-blocking sets in $\operatorname{PG}(n,q)$, where we set $s=2$ and $t=1$. In $\operatorname{PG}(n,q)$, an $(s,t)$-blocking set refers to a set of $t$-spaces such that each $s$-space is incident with at least one chosen $t$-space. This is a notoriously difficult problem, as it is equivalent to determining the size of certain $q$-Tur\'an designs and $q$-covering designs. We present an improvement on the upper bounds of Etzion and of Metsch via a refined scheme for a recursive construction, which in fact enables improvement in the general case as well.

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Benedek Kovács, Zoltán Lóránt Nagy, Dávid R. Szabó. 2024-10-10. Blocking Planes by Lines in $\operatorname{PG}(n,q)$. https://arxiv.org/abs/2410.07937

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