Uniformly cross intersecting families


Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be -cross-intersecting iff |AB|= for all AA and BB. Denote by P e (n) the maximum value of |A||B| over all such pairs. The best known upper bound on P e (n) is Θ(2n), by Frankl and Rödl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2, a simple construction of an -cross-intersecting pair (A,B) with |A||B| = \( \left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right) \)2n−2 = Θ(2n/\( \sqrt \ell \)), and conjectured that this is best possible. Consequently, Sgall asked whether or not P e (n) decreases with .

In this paper, we confirm the above conjecture of Ahlswede et al. for any sufficiently large , implying a positive answer to the above question of Sgall as well. By analyzing the linear spaces of the characteristic vectors of A,B over ℝ, we show that there exists some 0 > 0, such that P e (n) ≤ \( \left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right) \)2n−2 for all 0. Furthermore, we determine the precise structure of all the pairs of families which attain this maximum.

This is a preview of subscription content, access via your institution.


  1. [1]

    R. Ahlswede, N. Cai and Z. Zhang: A general 4-words inequality with consequences for 2-way communication complexity, Adv. in Appl. Math.10 (1989), 75–94.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    N. Alon and J. H. Spencer: The Probabilistic Method, Second Edition, Wiley, New York, 2000.

    Google Scholar 

  3. [3]

    L. Babai and P. Frankl: Linear Algebra Methods in Combinatorics, Preliminary Version 2, Dept. of Computer Science, The Univesity of Chicago, 1992.

  4. [4]

    P. Erdős: On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. (2nd ser.)51 (1945), 898–902.

    Article  Google Scholar 

  5. [5]

    P. Erdős: Problems and results in graph theory and combinatorial analysis, in: Proc. of the Fifth British Comb. Conf. 1975 Aberdeen, pp. 169–192, Congressus Numerantium, No. XV, Utilitas Math., Winnipeg, Man., 1976.

    Google Scholar 

  6. [6]

    P. Erdős, C. Ko and R. Rado: Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 212 (1961), 313–320.

    Article  MathSciNet  Google Scholar 

  7. [7]

    P. Frankl: Extremal set systems, in: R. L. Graham, M. Grötschel, L. Lovász (eds.), Handbook of Combinatorics, Vol. 1, 2, pp. 1293–1329, Elsevier, Amsterdam, 1995.

    Google Scholar 

  8. [8]

    P. Frankl and Z. Füredi: Forbidding just one intersection, J. Combin. Theory Ser. A39(2) (1985), 160–176.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    P. Frankl and V. Rödl: Forbidden intersections, Trans. Amer. Math. Soc. 300 (1987), 259–286.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    P. Frankl and R. M. Wilson: Intersection theorems with geometric consequences, Combinatorica1(4) (1981), 357–368.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    G. Katona: Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar. 15 (1964), 329–337.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    P. Keevash and B. Sudakov: On a restricted cross-intersection problem, J. Combinatorial Theory Ser. A113 (2006), 1536–1542.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    J. Littlewood and C. Offord: On the number of real roots of a random algebraic equation III, Mat. Sbornik12 (1943), 277–285.

    MathSciNet  Google Scholar 

  14. [14]

    D. Lubell: A short proof of Sperner’s theorem, Journal of Combinatorial Theory1 (1966), 299.

    Article  MathSciNet  Google Scholar 

  15. [15]

    D. K. Ray-Chaudhuri and R. M. Wilson: On t-designs, Osaka J. Math. 12 (1975), 737–744.

    MATH  MathSciNet  Google Scholar 

  16. [16]

    H. Robbins: A remark on Stirling’s formula, Amer. Math. Monthly62 (1955), 26–29.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    J. Sgall: Bounds on pairs of families with restricted intersections, Combinatorica19(4) (1999), 555–566.

    MATH  Article  MathSciNet  Google Scholar 

  18. [18]

    E. Sperner: Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928), 544–548.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Noga Alon.

Additional information

Research supported in part by the Israel Science Foundation, by a USA-Israel BSF grant, by an ERC advanced grant, by NSF grant CCF 0832797 and by the Ambrose Monell Foundation.

Research partially supported by a Charles Clore Foundation Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alon, N., Lubetzky, E. Uniformly cross intersecting families. Combinatorica 29, 389–431 (2009). https://doi.org/10.1007/s00493-009-2332-6

Download citation

Mathematics Subject Classification (2000)

  • 05D05