Journal of Geometry

, Volume 101, Issue 1–2, pp 31–50 | Cite as

On q-analogues and stability theorems

  • Aart Blokhuis
  • Andries BrouwerEmail author
  • Tamás Szőnyi
  • Zsuzsa Weiner
Open Access


In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.

Mathematics Subject Classification (2010)

05D05 05B25 


q-analogues Erdős–Ko–Rado theorem q-Kneser graph chromatic number blocking set stability results 


Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Bárány I.: A short proof of Kneser’s conjecture. J. Combin. Theory, Ser. A 25, 325–326 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bey C.: Polynomial LYM inequalities. Combinatorica 25, 19–38 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bose R.C., Burton R.C.: A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes. J. Combin. Theory 1, 96–104 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Blokhuis A.: On the size of a blocking set in PG(2,p). Combinatorica 14, 273–276 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blokhuis, A.: Blocking sets in Desarguesian planes. In: Miklós, D., et al. (eds.) Paul Erdős is Eighty, vol. 2, pp. 133–155. Bolyai Soc. Math. Studies, Budapest (1996)Google Scholar
  6. 6.
    Blokhuis A., Brouwer A.E.: Blocking sets in Desarguesian projective planes. Bull. London Math. Soc. 18, 132–134 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Blokhuis A., Brouwer A.E., Szőnyi T.: Covering all points except one. J. Algebraic Combin., 32, 59–66 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Blokhuis A., Brouwer A.E., Chowdhury A., Frankl P., Patkós B., Mussche T., Szőnyi T.: A Hilton–Milner theorem for vector spaces. Electron. J. Combin. 17, R71 (2010)Google Scholar
  9. 9.
    Blokhuis, A., Brouwer, A.E., Szőnyi, T.: On the chromatic number of q-Kneser graphs. Des. Codes Cryptogr. (to appear)Google Scholar
  10. 10.
    Boros E., Szőnyi T.: On the sharpness of the theorem of B. Segre. Combinatorica 6, 261–268 (1986)zbMATHCrossRefGoogle Scholar
  11. 11.
    Brouwer, A.E., Draisma, J., Güven, Ç: The unique coclique extension property for apartments in buildings (2011, manuscript)Google Scholar
  12. 12.
    Brouwer A.E., Schrijver A.: The blocking number of an affine space. J. Combin. Theory Ser. A 24, 251–253 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bruen A.A.: Baer subplanes and blocking sets. Bull. Am. Math. Soc. 76, 342–344 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chowdhury A., Godsil C., Royle G.: Colouring lines in a projective space. J. Combin. Theory Ser. A 113, 228–236 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chowdhury A., Patkós B.: Shadows and intersection theorems in vector spaces. J. Combin. Theory Ser. A 117, 1095–1106 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Deza M., Frankl P.: Erdős–Ko–Rado theorem – 22 years later. SIAM J. Algebr. Discret. Methods 4, 419–431 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Dodunekov S., Storme L., Van de Voorde G.: Partial covers of PG(n,q). Eur. J. Combin. 31, 1611–1616 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ebert G.: Partitioning projective geometries into caps. Can. J. Math. 37, 1163–1175 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Eisfeld J., Storme L., Sziklai P.: Minimal covers of the Klein quadric. J. Combin. Theory Ser. A 95, 145–157 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ellis, D: Non-trivial t-intersecting families of subspaces (2009, manuscript)Google Scholar
  21. 21.
    Erdős P.: Some theorems on graphs. Riveon Lematematika 9, 13–17 (1955)MathSciNetGoogle Scholar
  22. 22.
    Erdős P.: On a theorem of Rademacher-Turán. Ill. J. Math. 6, 122–127 (1962)Google Scholar
  23. 23.
    Erdős P.: On the number of triangles contained in certain graphs. Can. Math. Bull. 7, 53–56 (1974)CrossRefGoogle Scholar
  24. 24.
    Erdős P., Ko C., Rado R.: Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 12(2), 313–320 (1961)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions In: Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), vol. II, Colloq. Math. Soc. János Bolyai, Amsterdam, vol. 10, pp. 609–627 (1975)Google Scholar
  26. 26.
    Erdős, P., Győri, E., Simonovits, M.: How many edges should be deleted to make a triangle-free graph bipartite. In: Halász, G. et al. (eds.) Sets, Graphs and Numbers. vol. 60, pp. 239–263. Colloq. Math. Soc. J. Bolyai, Amsterdam (1991)Google Scholar
  27. 27.
    Fisher J.C., Hirschfeld J.W.P., Thas J.A.: Complete arcs on planes of square order. Ann. Discret. Math. 30, 243–250 (1986)MathSciNetGoogle Scholar
  28. 28.
    Frankl P.: On Sperner families satisfying an additional condition. J. Combin. Theory Ser. A 20, 1–11 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Frankl, P.: The shifting technique in extremal set theory. In: Surveys in combinatorics 1987 (New Cross, 1987), London Math. Soc. Lecture Note Ser, vol. 123, pp. 81–110. Cambridge University Press, Cambridge (1987)Google Scholar
  30. 30.
    Frankl P., Wilson R.M.: The Erdős–Ko–Rado theorem for vector spaces. J. Combin. Theory Ser. A 43, 228–236 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Gács A., Szőnyi T., Weiner Zs.: On the spectrum of minimal blocking sets. J. Geom. 76, 256–281 (2003)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Gerbner D., Patkós B.: Profile vectors in the lattice of subspaces. Discret. Math. 309, 2861–2869 (2009)zbMATHCrossRefGoogle Scholar
  33. 33.
    Godsil C.D., Newman M.W.: Independent sets in association schemes. Combinatorica 26, 431–443 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Greene, C., Kleitman, D.J.: Proof techniques in the theory of finite sets. In: Studies in combinatorics, MAA Stud. Math. vol. 17, pp. 22–79. Math. Assoc. America, Washington, D.C. (1978)Google Scholar
  35. 35.
    Greene J.E.: A new short proof of Kneser’s conjecture. Am. Math. Monthly 109, 918–920 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Hanson D., Toft B.: k-saturated graphs of chromatic number at least k. Ars Combin. 31, 159–164 (1991)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Harrach, N.V., Storme, L.: Partial covers of PG(n,q) (2011, manuscript)Google Scholar
  38. 38.
    Hilton A.J.W., Milner E.C.: Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 18(2), 369–384 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Hirschfeld J.W.P.: The 1959 Annali di Matematica paper of Beniamino Segre and its legacy. J. Geom. 76, 82–94 (2003)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hirschfeld J.W.P., Korchmáros G.: Arcs and curves over a finite field. Finite Fields Appl. 5, 393–408 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Hsieh W.N.: Intersection theorems for systems of finite vector spaces. Discret. Math. 12, 1–16 (1975)zbMATHCrossRefGoogle Scholar
  42. 42.
    Jamison R.E.: Covering finite fields with cosets of subspaces. J. Combin. Theory Ser. A 22, 253–266 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Lovász L.: Kneser’s conjecture, chromatic number, and homotopy. J. Combin. Theory Ser. A 25, 319–324 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Lovász, L., Simonovits, M.: On the number of complete subgraphs of a graph. In: Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), Congressus Numerantium XV, pp. 431–441 (1976)Google Scholar
  45. 45.
    Lovász, L., Simonovits, M.: On the number of complete subgraphs of a graph II. In: Studies in pure mathematics, pp. 459–495. Birkhäuser, Basel (1983)Google Scholar
  46. 46.
    Matoușek J.: A combinatorial proof of Kneser’s conjecture. Combinatorica 24, 163–170 (2004)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Metsch K.: How many s-subspaces must miss a point set in PG(d,q). J. Geom. 86, 154–164 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Newman, M.: Independent Sets and Eigenspaces. Ph.D. Thesis, University of Waterloo (2004)Google Scholar
  49. 49.
    Polverino O.: Small blocking sets in PG(2,p 3). Des. Codes Cryptogr. 20, 319–324 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Segre B.: Le geometrie di Galois. Ann. Mat. Pura Appl. (4) 48, 1–96 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Segre B.: Introduction to Galois geometries. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I. (8) 8, 133–236 (1967)MathSciNetGoogle Scholar
  52. 52.
    Simonovits, M.: Some of my Favorite Erdős Theorems and Related Results, Theories. In: G. Halász, G. et al. (eds.) Paul Erdős and His Mathematics, vol. II, pp. 565–635. Springer, Heidelberg (2002)Google Scholar
  53. 53.
    Sziklai P.: On small blocking sets and their linearity. J. Combin. Theory Ser A. 115, 1167–1182 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Sziklai P., Szőnyi T.: Blocking sets and algebraic curves. Rend. Circ. Mat. Palermo 51, 71–86 (1998)Google Scholar
  55. 55.
    Szőnyi T.: Blocking sets in Desarguesian affine and projective planes. Finite Fields Appl. 3, 187–202 (1997)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Szőnyi T.: Around Rédei’s theorem. Discret. Math. 208/9, 557–575 (1999)CrossRefGoogle Scholar
  57. 57.
    Szőnyi, T., Weiner, Zs.: On stability results in finite geometry. (preprint)
  58. 58.
    Szőnyi, T., Weiner, Zs.: A stability theorem for lines in Galois planes of prime order. Des. Codes Cryptogr. (to appear)Google Scholar
  59. 59.
    Tanaka H.: Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs. J. Combin. Theory Ser. A 113, 903–910 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Tanaka, H.: Vertex subsets with minimal width and dual width in Q-polynomial distance regular graphs. arXiv:1011.2000 (Preprint)Google Scholar
  61. 61.
    Tanaka, H.: The Erdős–Ko–Rado theorem for twisted Grassmann graphs. arXiv:1012.5692 (Preprint)Google Scholar
  62. 62.
    Thas J.A.: Complete arcs and algebraic curves in PG(2,q). J. Algebra 106, 451–464 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Tokushige N.: Multiply-intersecting families revisited. J. Combin. Theory Ser. B 97, 929–948 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Turán P.: On an extremal problem in graph theory (in Hungarian). Matematikai és Fizikai Lapok 48, 436–452 (1941)zbMATHGoogle Scholar
  65. 65.
    Voloch J.F.: Arcs in projective planes over prime fields. J. Geom. 38, 198–200 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Voloch, J.F.: Complete arcs in Galois planes of non-square order. In: Hirschfeld, J.W.P., et al. (eds.) Advances in Finite Geometries and Designs, pp. 401–406. Oxford University Press, Oxford (1991)Google Scholar
  67. 67.
    Weiner Zs., Szőnyi T.: Proof of a conjecture of Metsch. J. Combin. Theory Ser. A 118, 2066–2070 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Aart Blokhuis
    • 1
  • Andries Brouwer
    • 1
    Email author
  • Tamás Szőnyi
    • 2
    • 3
  • Zsuzsa Weiner
    • 2
    • 4
  1. 1.Department of MathematicsEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Department of Computer ScienceEötvös Loránd UniversityBudapestHungary
  3. 3.Computer and Automation Research Institute of the Hungarian Academy of SciencesBudapestHungary
  4. 4.Prezi.comBudapestHungary

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