An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
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Klaus Metsch
Abstract
In 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.
Acknowledgements
The author would like to thank Bernhard Mühlherr for many discussions on the subject. The author would also like to thank the referee for many valuable suggestions that improved this presentation.
References
[1] A. Blokhuis, A. E. Brouwer and C. Güven, Cocliques in the Kneser graph on the point-hyperplane flags of a projective space, Combinatorica 34 (2014), no. 1, 1–10. 10.1007/s00493-014-2779-ySearch in Google Scholar
[2] P. Erdős, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1961), 313–320. 10.1093/qmath/12.1.313Search in Google Scholar
[3] P. Frankl and R. M. Wilson, The Erdős–Ko–Rado theorem for vector spaces, J. Combin. Theory Ser. A 43 (1986), no. 2, 228–236. 10.1016/0097-3165(86)90063-4Search in Google Scholar
[4] C. D. Godsil and M. W. Newman, Independent sets in association schemes, Combinatorica 26 (2006), no. 4, 431–443. 10.1007/s00493-006-0024-zSearch in Google Scholar
[5] J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Oxford Math. Monogr., Clarendon Press, New York, 1991. Search in Google Scholar
[6] W. N. Hsieh, Intersection theorems for systems of finite vector spaces, Discrete Math. 12 (1975), 1–16. 10.1016/0012-365X(75)90091-6Search in Google Scholar
[7]
F. Ihringer, K. Metsch and B. Mühlherr,
An EKR theorem for finite buildings of type
[8] K. Metsch, An Erdős–Ko–Rado theorem for finite classical polar spaces, J. Algebraic Combin. 43 (2016), no. 2, 375–397. 10.1007/s10801-015-0637-7Search in Google Scholar
[9] K. Metsch, A note on Erdős–Ko–Rado sets of generators in Hermitian polar spaces, Adv. Math. Commun. 10 (2016), no. 3, 541–545. 10.3934/amc.2016024Search in Google Scholar
[10] V. Pepe, L. Storme and F. Vanhove, Theorems of Erdős–Ko–Rado type in polar spaces, J. Combin. Theory Ser. A 118 (2011), no. 4, 1291–1312. 10.1016/j.jcta.2011.01.003Search in Google Scholar
[11] D. Stanton, Some Erdős–Ko–Rado theorems for Chevalley groups, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 160–163. 10.1137/0601019Search in Google Scholar
[12] H. Tanaka, Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs, J. Combin. Theory Ser. A 113 (2006), no. 5, 903–910. 10.1016/j.jcta.2005.08.006Search in Google Scholar
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- Regularity of symbolic powers and arboricity of matroids
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- An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
- Rankin–Selberg gamma factors of level zero representations of GLn
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Articles in the same Issue
- Frontmatter
- Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
- Infinite families of equivariantly formal toric orbifolds
- Bounds for GL3L-functions in depth aspect
- Nonlinear Dirichlet problems with unilateral growth on the reaction
- K-invariant cusp forms for reductive symmetric spaces of split rank one
- Cellularity in quotient spaces of topological groups
- Shifted convolution sums for higher rank groups
- Dimension quotients, Fox subgroups and limits of functors
- Large sums of Hecke eigenvalues of holomorphic cusp forms
- K-theory classification of graded ultramatricial algebras with involution
- Regularity of symbolic powers and arboricity of matroids
- On some problems concerning symmetrization operators
- An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
- Rankin–Selberg gamma factors of level zero representations of GLn
- Sobolev’s inequality for double phase functionals with variable exponents
- Independence of Artin L-functions
- Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
