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Journal of Algebraic Combinatorics

, Volume 47, Issue 4, pp 529–541 | Cite as

An EKR-theorem for finite buildings of type \(D_{\ell }\)

  • Ferdinand Ihringer
  • Klaus Metsch
  • Bernhard Mühlherr
Article
  • 135 Downloads

Abstract

We define optimal EKR-sets in finite buildings. This definition is motivated by various contributions on optimal EKR-sets in finite projective spaces and polar spaces. Our main result is the classification of optimal EKR-sets of type \(\{ \ell , \ell -1\}\) in finite building of type \(D_{\ell }\) with \(\ell \) even. As it is the case for most of the known optimal EKR-sets in finite buildings, our EKR-sets have a natural center. This provides some evidence that the EKR-problem for finite buildings and Tits center conjecture are closely related.

Keywords

EKR-sets Polar spaces Finite buildings 

Mathematics Subject Classification

05C35 51A50 51E24 

Notes

Acknowledgements

We would like to thank the anonymous referee for reading our manuscript very carefully.

References

  1. 1.
    Blokhuis, A., Brouwer, A.E., Szőnyi, T.: Maximal cocliques in the Kneser graph on point–plane flags in \({\rm PG}(4,q)\). Eur. J. Comb. 35, 95–104 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blokhuis, A., Brouwer, A.E.: Cocliques in the Kneser graph on line-plane flags in \({\rm PG}(4, q)\). Combinatorica, AcceptedGoogle Scholar
  3. 3.
    Blokhuis, A., Brouwer, A.E., Güven, Çiçek: Cocliques in the Kneser graph on the point-hyperplane flags of a projective space. Combinatorica 34(1), 1–10 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris, (1968)Google Scholar
  5. 5.
    Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-regular graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1989)Google Scholar
  6. 6.
    Cameron, P.J., Ku, C.Y.: Intersecting families of permutations. Eur. J. Comb. 24(7), 881–890 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ellis, D., Friedgut, E., Pilpel, H.: Intersecting families of permutations. J. Am. Math. Soc. 24(3), 649–682 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2(12), 313–320 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Filmus, Y.: Spectral Methods in Extremal Combinatorics. PhD thesis, University of Toronto, (2013)Google Scholar
  10. 10.
    Frankl, P., Wilson, R.M.: The Erdős-Ko-Rado theorem for vector spaces. J. Comb. Theory Ser. A 43(2), 228–236 (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    Godsil, C., Meagher, K.: Erdős-Ko-Rado Theorems: Algebraic Approaches. Number 149 in Cambridge Studies in Advanced Mathematics. Cambridge University Press (2016)Google Scholar
  12. 12.
    Hsieh, W.N.: Intersection theorems for systems of finite vector spaces. Discrete Math. 12, 1–16 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hurlbert, G., Kamat, V.: Erdős-Ko-Rado theorems for chordal graphs and trees. J. Combin. Theory Ser. A 118(3), 829–841 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ihringer, F., Metsch, K.: On the maximum size of Erdős-Ko-Rado sets in \(H(2d+1, q^2)\). Des. Codes Cryptogr. 72(2), 311–316 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Leeb, B., Ramos-Cuevas, C.: The center conjecture for spherical buildings of types \(F_4\) and \(E_6\). Geom. Funct. Anal. 21(3), 525–559 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Meagher, K., Moura, L., Stevens, B.: A Sperner-type theorem for set-partition systems. Electron. J. Comb. 12(Note 20), 6 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Metsch, K.: An Erdős-Ko-Rado theorem for finite classical polar spaces. J. Algebraic Comb. 43(2), 375–397 (2016)CrossRefzbMATHGoogle Scholar
  18. 18.
    Mühlherr, B., Tits, J.: The center conjecture for non-exceptional buildings. J. Algebra 300(2), 687–706 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Newman, M.W.: Independent Sets and Eigenspaces. PhD thesis, University of Waterloo, Waterloo, Canada, (2004)Google Scholar
  20. 20.
    Pepe, V., Storme, L., Vanhove, F.: Theorems of Erdős-Ko-Rado type in polar spaces. J. Comb. Theory Ser. A 118(4), 1291–1312 (2011)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ramos-Cuevas, C.: The center conjecture for thick spherical buildings. Geom. Dedic. 166, 349–407 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Serre, J.: Complète réductibilité. Astérisque, (299):Exp. No. 932, viii, 195–217 . Séminaire Bourbaki. Vol. 2003/2004 (2005)Google Scholar
  23. 23.
    Tanaka, H.: Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs. J. Comb. Theory Ser. A 113(5), 903–910 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tits, J.: Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, vol. 386. Springer, Berlin (1974)Google Scholar
  25. 25.
    Vanhove, F.: Incidence geometry from an algebraic graph theory point of view. PhD thesis, Ghent University, (2011)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsThe Hebrew University of JerusalemGivat Ram, JerusalemIsrael
  2. 2.Mathematisches InstitutJustus-Liebig-UniversitätGiessenGermany

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