On the densities of cliques and independent sets in graphs

Abstract

Let r; s≥2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r = 2 or s = 2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.

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

References

  1. [1]

    M.H. Albert, M.D. Atkinson, C.C. Handley, D.A. Holton and W. Stromquist: On packing densities of permutations, Electron. J. Combin. 9 (2002), #R5.

  2. [2]

    V. Chvátal and P. Hammer: Aggregation of inequalities in integer programming, Ann. Discrete Math. 1 (1977), 145–162.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    S. Das, H. Huang, J. Ma, H. Naves and B. Sudakov: A problem of Erdős on the minimum of k-cliques, J. Combinatorial Theory Ser. B 103 (2013), 344–373.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    P. Erdős: On the number of complete subgraphs contained in certain graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 (1962), 459–464.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    P. Erdős and A. Stone: On the structure of linear graphs, Bull. Am. Math. Soc. 52 (1946), 1087–1091.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    P. Frankl: The shifting techniques in extremal set theory, in: Surveys in Combinatorics, Lond. Math. Soc. Lect. Note Ser. 123 (1987), 81–110.

    MathSciNet  Google Scholar 

  7. [7]

    P. Frankl, M. Kato, G. Katona and N. Tokushige: Two-colorings with many monochromatic cliques in both colors, J. Comb. Theory Ser. B 103 (2013), 415–427.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    A. Goodman: On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959), 778–783.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    G. Katona: A theorem of finite sets, in: Theory of Graphs Akadémia Kiadó, Budapest (1968), 187–207.

    Google Scholar 

  10. [10]

    P. Keevash: Shadows and intersections: stability and new proofs, Adv. Math. 218 (2008), 1685–1703.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    J. Kruskal: The number of simplicies in a complex, Mathematical Optimization Techniques, Univ. of California Press (1963), 251–278.

    Google Scholar 

  12. [12]

    V. Nikiforov: The number of cliques in graphs of given order and size, Trans. Amer. Math. Soc. 363 (2011), 1599–1618.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    O. Pikhurko and E.R. Vaughan: Minimum number of k-cliques in graphs with bounded independence number, Combin. Probab. Computing 22 (2013), 910–934.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    A. Razborov: On the minimal density of triangles in graphs, Combin. Probab. Computing 17 (2008), 603–618.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    C. Reiher: Minimizing the number of cliques in graphs of given order and edge density, manuscript.

  16. [16]

    A. Thomason: A disproof of a conjecture of Erdős in Ramsey theory, J. London Math. Soc. (2) 39(2) (1989), 246–255.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    P. Turán: On an extremal problem in graph theory, Matematikai ás Fizikai Lapok 48 (1941), 436–452.

    Google Scholar 

  18. [18]

    A. Zykov: On some properties of linear complexes, Mat. Sbornik N.S. 24 (1949), 163–188.

    MathSciNet  Google Scholar 

  19. [19]

    F. Franek and V. Rödl: 2-colorings of complete graphs with small number of monochromatic K 4 subgraphs, Discrete Mathematics 114 (1993), 199–203.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hao Huang.

Additional information

Research supported in part by NSF grant DMS-1128155.

Research supported in part by the Israel Science Foundation and by a USA-Israel BSF grant.

Research supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huang, H., Linial, N., Naves, H. et al. On the densities of cliques and independent sets in graphs. Combinatorica 36, 493–512 (2016). https://doi.org/10.1007/s00493-015-3051-9

Download citation

Mathematics Subject Classification (2000)

  • 05C35
  • 05C69
  • 05D05