Advertisement

Designs, Codes and Cryptography

, Volume 65, Issue 3, pp 233–245 | Cite as

The de Bruijn–Erdős theorem for hypergraphs

  • Noga Alon
  • Keith E. Mellinger
  • Dhruv Mubayi
  • Jacques Verstraëte
Article

Abstract

Fix integers nr ≥ 2. A clique partition of \({{[n] \choose r}}\) is a collection of proper subsets \({A_1, A_2, \ldots, A_t \subset [n]}\) such that \({\bigcup_i{A_i \choose r}}\) is a partition of \({{[n]\choose r}}\) . Let cp(n, r) denote the minimum size of a clique partition of \({{[n] \choose r}}\) . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3,
$${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$
We conjecture cp(n, r) = (1 + o(1))n r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman–Mullin–Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1 + o(1))n r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n r ) of the r-sets of [n] are covered. We also give an absolute lower bound \({{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}\) when nq 2 + qr − 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.

Keywords

Hypergraph de Bruijn-Erdős Zarankiewicz problem 

Mathematics Subject Classification (2000)

05B05 05C65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon N.: Decomposition of the complete r-graph into complete r-partite r-graphs. Graph. Combinator 2, 95–100 (1986)zbMATHCrossRefGoogle Scholar
  2. 2.
    Alon N., Rónyai L., Szabó T.: Norm-graphs: variations and applications. J. Combin. Theory Ser. B 76(2), 280–290 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barlotti A.: Un’ estensione del teorema di Segre-Kustaanheimo. Boll. U.M.I 10, 498–506 (1955)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Benz W.: Vorlesungen ber Geometrie der Algebren, Geometrien von Mbius, Laguerre-Lie, Minkowski in einheitlicher und grundlagengeometrischer Behandlung. Die Grundlehren der mathematischen Wissenschaften, Band 197. Springer, Berlin (1973).Google Scholar
  5. 5.
    Cioaba S., Kündgen A., Verstraëte J.: On decompositions of complete hypergraphs. J. Combin. Theory A 116, 1232–1234 (2009)zbMATHCrossRefGoogle Scholar
  6. 6.
    Colbourn C., Mathon R.: Steiner systems. In: Handbook of Combinatorial Designs, 2nd edn., pp. 102–110. Chapman & Hall/CRC, Boca Raton, FL (2007).Google Scholar
  7. 7.
    de Bruijn N.G., Erdős P.: On a combinatorial problem. Nederl. Akad. Wetensch. Proc. 51, (1948) 1277–1279 = Indagationes Math 10, 421–423 (1948)Google Scholar
  8. 8.
    Dembowski P.: Inversive planes of even order. Bull. Am. Math. Soc 69, 850–854 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dembowski P., Hughes D.R.: On finite inversive planes. J. London Math. Soc 40, 171–182 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fisher R.A.: An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugenic 10, 52–75 (1940)Google Scholar
  11. 11.
    Fowler J.: A short proof of Totten’s classification of restricted linear spaces. Geometriae Dedicata 15(4), 413–422 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Füredi Z.: New asymptotics for bipartite Turán numbers. J. Combin. Theory Ser. A 75(1), 141–144 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Graham R.L., Pollak H.O.: On the addressing problem for loop switching. Bell System Tech 680 J 50, 2495–2519 (1971)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hardy G.H., Littlewood J.E., Polya G.: Inequalities, 2nd edn., pp. 43–44. Cambridge University Press, Cambridge, England (1988)zbMATHGoogle Scholar
  15. 15.
    Hartman A., Mullin R.C., Stinson D.: Exact covering configurations and Steiner Systems. J. London. Math. Soc 25(2), 193–200 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hirschfeld J.: Finite projective spaces of three dimensions, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1985)Google Scholar
  17. 17.
    Hirschfeld J.: Projective geometries over finite fields 2nd edn Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1998)Google Scholar
  18. 18.
    Kővári T., Sós V. T., Turán P.: On a problem of K Zarankiewicz. Colloquium Math 3, 50–57 (1954)MathSciNetGoogle Scholar
  19. 19.
    Lam C.: The search for a finite projective plane of order 10 Amer. Math. Monthly 98(4), 305–318 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Ray-Chaudhuri D.K., Wilson R.M.: On t-designs. Osaka J. Math 12, 737–744 (1975)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Segre B.: Ovals in a finite projective plane. Canad. J. Math 7, 414–416 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Thas J.: The affine plane AG(2, q), q odd, has a unique one point extension. Invent. math 118, 133–139 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Tits J.: Ovoïdes et groupes de Suzuki. Arch. Math 13, 187–198 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Totten J.: Classification of restricted linear spaces. Canad. J. Math 28, 321–333 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    van Lint R., Wilson R.: Designs, Graphs, Codes, and Their Links. Cambridge University Press (1991).Google Scholar
  26. 26.
    Vishwanathan S.: A counting proof of the Graham-Pollak Theorem. preprint.Google Scholar
  27. 27.
    Wilker J.B.: Inversive geometry. The geometric vein, pp. 379–442. Springer, New York-Berlin (1981).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Noga Alon
    • 1
  • Keith E. Mellinger
    • 2
  • Dhruv Mubayi
    • 3
  • Jacques Verstraëte
    • 4
  1. 1.Schools of Mathematics and Computer Science, Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsUniversity of Mary WashingtonFredericksburgUSA
  3. 3.Department of Mathematics, Statistics, and Computer ScienceUniversity of IllinoisChicagoUSA
  4. 4.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

Personalised recommendations