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Combinatorica

, Volume 29, Issue 4, pp 467–501 | Cite as

Generalizations of the removal lemma

  • Vojtěch Rödl
  • Mathias Schacht
Article

Abstract

Ruzsa and Szemerédi established the triangle removal lemma by proving that: For every η>0 there exists c>0 so that every sufficiently large graph on n vertices, which contains at most cn 3 triangles can be made triangle free by removal of at most η \( \left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right) \) edges. More general statements of that type regarding graphs were successively proved by several authors. In particular, Alon and Shapira obtained a generalization (which extends all the previous results of this type), where the triangle is replaced by a possibly infinite family of graphs and containment is induced.

In this paper we prove the corresponding result for k-uniform hypergraphs and show that: For every family ℱ of k-uniform hypergraphs and every η>0 there exist constants c > 0 and C > 0 such that every sufficiently large k-uniform hypergraph on n vertices, which contains at most cn νF induced copies of any hypergraph F ∈ ℱ on ν F ≤ C vertices can be changed by adding and deleting at most η \( \left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right) \) edges in such a way that it contains no induced copy of any member of ℱ.

Mathematics Subject Classification (2000)

05C35 05C65 68W20 

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References

  1. [1]
    N. Alon, E. Fischer, M. Krivelevich and M. Szegedy: Efficient testing of large graphs, Combinatorica20(4) (2000), 451–476.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    N. Alon and A. Shapira: A characterization of the (natural) graph properties testable with one-sided error, in: Proceedings of the fourty-sixth annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 2005, pp. 429–438.Google Scholar
  3. [3]
    N. Alon and A. Shapira: Every monotone graph property is testable, in: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing (New York, NY, USA), ACM Press, 2005, pp. 128–137.Google Scholar
  4. [4]
    C. Avart, V. Rödl and M. Schacht: Every monotone 3-graph property is testable, SIAM J. Discrete Math. 21(1) (2007), 73–92. (electronic).zbMATHMathSciNetGoogle Scholar
  5. [5]
    B. Bollobás, P. Erdős, M. Simonovits and E. Szemerédi: Extremal graphs without large forbidden subgraphs, Ann. Discrete Math. 3 (1978), 29–41, Advances in graph theory (Cambridge Combinatorial Conf., Trinity Coll., Cambridge, 1977).Google Scholar
  6. [6]
    C. Borgs, J. Chayes, L. Lovász, V. T. Sós, B. Szegedy and K. Vesztergombi: Graph limits and parameter testing, in: STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (New York), ACM, 2006, pp. 261–270.Google Scholar
  7. [7]
    W. G. Brown, P. Erdős and V. T. Sós: Some extremal problems on r-graphs, in: New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), Academic Press, New York, 1973, pp. 53–63.Google Scholar
  8. [8]
    R. A. Duke and V. Rödl: On graphs with small subgraphs of large chromatic number, Graphs Combin. 1(1) (1985), 91–96.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    P. Erdős: Problems and results on graphs and hypergraphs: similarities and differences; in: Mathematics of Ramsey theory (J. Nešetřil and V. Rödl, eds.), Algorithms Combin., vol. 5, Springer, Berlin, 1990, pp. 12–28.Google Scholar
  10. [10]
    P. Erdős, P. Frankl and V. Rödl: The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Combin. 2(2) (1986), 113–121.CrossRefMathSciNetGoogle Scholar
  11. [11]
    P. Frankl and V. Rödl: Extremal problems on set systems, Random Structures Algorithms20(2) (2002), 131–164.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    H. Furstenberg and Y. Katznelson: An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    H. Furstenberg and Y. Katznelson: An ergodic Szemerédi theorem for IPsystems and combinatorial theory, J. Analyse Math. 45 (1985), 117–168.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    O. Goldreich, S. Goldwasser and D. Ron: Property testing and its connection to learning and approximation, J. ACM45(4) (1998), 653–750.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    W. T. Gowers: Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. of Math. (2)166(3) (2007), 897–946.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Y. Kohayakawa, B. Nagle and V. Rödl: Efficient testing of hypergraphs (extended abstract), in: Automata, languages and programming, Lecture Notes in Comput. Sci., vol. 2380, Springer, Berlin, 2002, pp. 1017–1028.CrossRefGoogle Scholar
  17. [17]
    J. Komlós, A. Shokoufandeh, M. Simonovits and E. Szemerédi: The regularity lemma and its applications in graph theory, in: Theoretical aspects of computer science (Tehran, 2000), Lecture Notes in Comput. Sci., vol. 2292, Springer, Berlin, 2002, pp. 84–112.CrossRefGoogle Scholar
  18. [18]
    L. Lovász and B. Szegedy: Limits of dense graph sequences, Tech. Report MSRTR-2004-79, Microsoft Research, 2004.Google Scholar
  19. [19]
    L. Lovász and B. Szegedy: Graph limits and testing hereditary graph properties, Tech. Report MSR-TR-2005-110, Microsoft Research, 2005.Google Scholar
  20. [20]
    B. Nagle, V. Rödl and M. Schacht: The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms28(2) (2006), 113–179.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    V. Rödl, B. Nagle, J. Skokan, M. Schacht and Y. Kohayakawa: The hypergraph regularity method and its applications, Proc. Natl. Acad. Sci. USA102(23) (2005), 8109–8113 (electronic).zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    V. Rödl and M. Schacht: Regular partitions of hypergraphs: Counting lemmas; Combin. Probab. Comput. 16(6) (2007), 887–901.zbMATHMathSciNetGoogle Scholar
  23. [23]
    V. Rödl and M. Schacht: Regular partitions of hypergraphs: Regularity lemmas; Combin. Probab. Comput. 16(6) (2007), 833–885.zbMATHMathSciNetGoogle Scholar
  24. [24]
    V. Rödl, M. Schacht, E. Tengan and N. Tokushige: Density theorems and extremal hypergraph problems, Israel J. Math. 152 (2006), 371–380.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    V. Rödl and J. Skokan: Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms25(1) (2004), 1–42.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    V. Rödl and J. Skokan: Applications of the regularity lemma for uniform hypergraphs, Random Structures Algorithms28(2) (2006), 180–194.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    I. Z. Ruzsa and E. Szemerédi: Triple systems with no six points carrying three triangles, in: Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, vol. 18, North-Holland, Amsterdam, 1978, pp. 939–945.Google Scholar
  28. [28]
    J. Solymosi: A note on a question of Erdős and Graham, Combin. Probab. Comput. 13(2) (2004), 263–267.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    J. Solymosi: Regularity, uniformity, and quasirandomness; Proc. Natl. Acad. Sci. USA102(23) (2005), 8075–8076 (electronic).zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    V. T. Sós, P. Erdős and W. G. Brown: On the existence of triangulated spheres in 3-graphs, and related problems; Period. Math. Hungar. 3(3–4) (1973), 221–228.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    E. Szemerédi: On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245, Collection of articles in memory of Juriĭ Vladimirovič Linnik.zbMATHMathSciNetGoogle Scholar
  32. [32]
    E. Szemerédi: Regular partitions of graphs, in: Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, vol. 260, CNRS, Paris, 1978, pp. 399–401.Google Scholar
  33. [33]
    T. Tao: A variant of the hypergraph removal lemma, J. Combin. Theory Ser. A113(7) (2006), 1257–1280.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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