Blocking Planes by Lines in $\operatorname{PG}(n,q)$
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arXiv
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| Format: | Preprint |
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2024
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| author | Kovács, Benedek Nagy, Zoltán Lóránt Szabó, Dávid R. |
| author_facet | Kovács, Benedek Nagy, Zoltán Lóránt Szabó, Dávid R. |
| contents | 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 $q$-Turán 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07937 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blocking Planes by Lines in $\operatorname{PG}(n,q)$ Kovács, Benedek Nagy, Zoltán Lóránt Szabó, Dávid R. Combinatorics 51E21 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 $q$-Turán 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. |
| title | Blocking Planes by Lines in $\operatorname{PG}(n,q)$ |
| topic | Combinatorics 51E21 |
| url | https://arxiv.org/abs/2410.07937 |