scholarly journals Cross-Intersecting Erdős-Ko-Rado Sets in Finite Classical Polar Spaces

10.37236/4734 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ferdinand Ihringer
Keyword(s):  
Maximum Size ◽  
Upper Bounds ◽  
Polar Space ◽  
Polar Spaces ◽  
The Cross ◽  
Classical Polar Spaces

A cross-intersecting Erdős-Ko-Rado set of generators of a finite classical polar space is a pair $(Y, Z)$ of sets of generators such that all $y \in Y$ and $z \in Z$ intersect in at least a point. We provide upper bounds on $|Y| \cdot |Z|$ and classify the cross-intersecting Erdős-Ko-Rado sets of maximum size with respect to $|Y| \cdot |Z|$ for all polar spaces except some Hermitian polar spaces.

An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces

2019 ◽  
Vol 31 (2) ◽  
pp. 491-502 ◽  
Author(s):  
Klaus Metsch
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Classical Polar Spaces

AbstractIn this paper, we call a set of lines of a finite classical polar space an Erdős–Ko–Rado set of lines if no two lines of the polar space are opposite, which means that for any two lines l and h in such a set there exists a point on l that is collinear with all points of h. We classify all largest such sets provided the order of the underlying field of the polar space is not too small compared to the rank of the polar space. The motivation for studying these sets comes from [7], where a general Erdős–Ko–Rado problem was formulated for finite buildings. The presented result provides one solution in finite classical polar spaces.

scholarly journals On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces

10.37236/90 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Hermitian Variety ◽  
Projective Embedding ◽  
Dual Polar Space ◽  
Transfinite Recursion ◽  
Symplectic Dual Polar Space ◽  
Dual Polar Spaces

Let $\Delta$ be a symplectic dual polar space $DW(2n-1,{\Bbb K})$ or a Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$, $n \geq 2$. We define a class of hyperplanes of $\Delta$ arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta$-Hermitian variety in ${\rm PG}(n-1,{\Bbb K})$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion.

scholarly journals On Local Packings of the Cross-Polytope

10.37236/8990 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Ji Hoon Chun
Keyword(s):  
Three Dimensional ◽  
Upper Bounds ◽  
The Cross

The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is related to packings. For the $n$-dimensional cross-polytope, we show that $2n$ points can be placed when $r\in\left(1-\frac{1}{n},1\right]$. For the three-dimensional cross-polytope, $10$ and $12$ points can be placed if and only if $r\in\left(\frac{3}{5},\frac{2}{3}\right]$ and $r\in\left(\frac{4}{7},\frac{3}{5}\right]$ respectively, and no more than $14$ points can be placed when $r\in\left(\frac{1}{2},\frac{4}{7}\right]$. Also, constructive arrangements of points that attain the upper bounds of $2n$, $10$, and $12$ are provided, as well as $13$ points for dimension $3$ when $r\in\left(\frac{1}{2},\frac{6}{11}\right]$.

scholarly journals Improved Bounds on mr(2,q) q=19,25,27

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Rumen Daskalov ◽  
Elena Metodieva
Keyword(s):  
Projective Plane ◽  
Maximum Size ◽  
Upper Bounds ◽  
Computer Search ◽  
Lower And Upper Bounds ◽  
Local Computer

An (n,r)-arc is a set of n points of a projective plane such that some r, but no r+1 of them, are collinear. The maximum size of an (n,r)-arc in PG(2, q) is denoted by mr(2, q). In this paper, a new (286, 16)-arc in PG(2,19), a new (341, 15)-arc, and a (388, 17)-arc in PG(2,25) are constructed, as well as a (394, 16)-arc, a (501, 20)-arc, and a (532, 21)-arc in PG(2,27). Tables with lower and upper bounds on mr(2, 25) and mr(2, 27) are presented as well. The results are obtained by nonexhaustive local computer search.

scholarly journals On the Smallest Non-Trivial Tight Sets in Hermitian Polar Spaces

10.37236/6461 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Jan De Beule ◽  
Klaus Metsch
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Tight Set ◽  
Tight Sets

We show that an $x$-tight set of the Hermitian polar spaces $\mathrm{H}(4,q^2)$ and $\mathrm{H}(6,q^2)$ respectively, is the union of $x$ disjoint generators of the polar space provided that $x$ is small compared to $q$. For $\mathrm{H}(4,q^2)$ we need the bound $x<q+1$ and we can show that this bound is sharp.

Classical Polar Spaces

2013 ◽  
pp. 447-497
Author(s):  
Francis Buekenhout ◽  
Arjeh M. Cohen
Keyword(s):  
Polar Spaces ◽  
Classical Polar Spaces

scholarly journals The Maximum Size of a Partial Spread in $H(4 n +1, q^2)$ is $q^{2 n +1}+1$

10.37236/251 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Frédéric Vanhove
Keyword(s):  
Maximum Size ◽  
Polar Space ◽  
Maximal Dimension ◽  
Partial Spread ◽  
Isotropic Subspaces ◽  
Hermitian Polar Space ◽  
Totally Isotropic Subspaces

We prove that in every finite Hermitian polar space of odd dimension and even maximal dimension $\rho$ of the totally isotropic subspaces, a partial spread has size at most $q^{\rho+1}+1$, where $GF(q^2)$ is the defining field. This bound is tight and is a generalisation of the result of De Beule and Metsch for the case $\rho=2$.

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