Incompatible Intersection Properties


Let F ⊂ 2 [n] be a family in which any three sets have non-empty intersection and any two sets have at least 32 elements in common. The nearly best possible bound F ≤ 2n−2 is proved. We believe that 32 can be replaced by 3 and provide a simple-looking conjecture that would imply this.

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


  1. [1]

    A. Brace and D. E. Daykin: A finite set covering theorem, Bulletin of the Australian Mathematical Society5 (1971), 197–202.

    Article  Google Scholar 

  2. [2]

    P. Erdős, C. Ko and R. Rado: Intersection theorems for systems of finite sets, The Quarterly Journal of Mathematics12 (1961), 313–320.

    MathSciNet  Article  Google Scholar 

  3. [3]

    P. Frankl: Families of finite sets satisfying an intersection condition, Bull. Austral. Math. Soc.15 (1976), 73–79.

    MathSciNet  Article  Google Scholar 

  4. [4]

    P. Frankl: The shifting technique in extremal set theory, Surveys in combinatorics123 (1987), 81–110.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    P. Frankl: Shadows and shifting, Graphs and Combinatorics7 (1991), 23–29.

    MathSciNet  Article  Google Scholar 

  6. [6]

    P. Frankl: Multiply-intersecting families, J. Comb. Theory Ser. B53 (1991), 195–234.

    MathSciNet  Article  Google Scholar 

  7. [7]

    P. Frankl: Improved bounds for Erdős’ Matching Conjecture, J. Comb. Theory Ser. A120 (2013), 1068–1072.

    Article  Google Scholar 

  8. [8]

    P. Frankl and A. Kupavskii: The Erdős Matching Conjecture and Concentration Inequalities, arXiv:1806.08855

  9. [9]

    P. Frankl and A. Kupavskii: Beyond the Erdős Matching Conjecture, arXiv:1901.09278

  10. [10]

    P. Frankl: Some exact results for multiply intersecting families, J. Comb. Theory Ser. B136 (2019), 222–248.

    MathSciNet  Article  Google Scholar 

  11. [11]

    T. E. Harris: A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Phil. Soc.56 (1960), 13–20.

    MathSciNet  Article  Google Scholar 

  12. [12]

    G. Kalai, N. Keller and Mossel: On the correlation of increasing families, J. Comb. Theory Ser. A144 (2016), 250–276.

    MathSciNet  Article  Google Scholar 

  13. [13]

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

    MathSciNet  Article  Google Scholar 

  14. [14]

    N. Keller, E. Mossel and A. Sen: Geometric influences II: Correlation inequalities and noise sensitivity, Ann. Inst. Henri Poincare50 (2014), 1121–1139.

    MathSciNet  Article  Google Scholar 

  15. [15]

    D. J. Kleitman: Families of Non-Disjoint Subsets, J. Combin. Theory1 (1966), 153–155.

    MathSciNet  Article  Google Scholar 

  16. [16]

    M. Talagrand: How much are increasing sets positively correlated?, Combinatorica16 (1996), 243–258.

    MathSciNet  Article  Google Scholar 

Download references


We thank Stijn Cambie for his remarks, presented in the previous section, and useful comments on the presentation of the paper. We also thank the anonymous referees for their helpful comments.

Author information



Corresponding author

Correspondence to Andrey Kupavskii.

Additional information

The research of the second author was supported by the EPSRC grant no. EP/N019504/1, the Russian Foundation for Basic Research (grant no. 18-01-00355) and the Council for the Support of Leading Scientific Schools of the President of the Russian Federation (grant no. N.Sh.-6760.2018.1).

A property is simply a class of families, and a family has that property if and only if it belongs to the class.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Frankl, P., Kupavskii, A. Incompatible Intersection Properties. Combinatorica 39, 1255–1266 (2019).

Download citation

Mathematics Subject Classification (2010)

  • 05D05