## 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*.

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

## References

- [1]
N. Alon, E. Fischer, M. Krivelevich and M. Szegedy: Efficient testing of large graphs,

*Combinatorica***20(4)**(2000), 451–476. - [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. - [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. - [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). - [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). - [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. - [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. - [8]
R. A. Duke and V. Rödl: On graphs with small subgraphs of large chromatic number,

*Graphs Combin*.**1(1)**(1985), 91–96. - [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. - [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. - [11]
P. Frankl and V. Rödl: Extremal problems on set systems,

*Random Structures Algorithms***20(2)**(2002), 131–164. - [12]
H. Furstenberg and Y. Katznelson: An ergodic Szemerédi theorem for commuting transformations,

*J. Analyse Math*.**34**(1978), 275–291. - [13]
H. Furstenberg and Y. Katznelson: An ergodic Szemerédi theorem for IPsystems and combinatorial theory,

*J. Analyse Math*.**45**(1985), 117–168. - [14]
O. Goldreich, S. Goldwasser and D. Ron: Property testing and its connection to learning and approximation,

*J. ACM***45(4)**(1998), 653–750. - [15]
W. T. Gowers: Hypergraph regularity and the multidimensional Szemerédi theorem,

*Ann. of Math. (2)***166(3)**(2007), 897–946. - [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. - [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. - [18]
L. Lovász and B. Szegedy:

*Limits of dense graph sequences*, Tech. Report MSRTR-2004-79, Microsoft Research, 2004. - [19]
L. Lovász and B. Szegedy:

*Graph limits and testing hereditary graph properties*, Tech. Report MSR-TR-2005-110, Microsoft Research, 2005. - [20]
B. Nagle, V. Rödl and M. Schacht: The counting lemma for regular

*k*-uniform hypergraphs,*Random Structures Algorithms***28(2)**(2006), 113–179. - [21]
V. Rödl, B. Nagle, J. Skokan, M. Schacht and Y. Kohayakawa: The hypergraph regularity method and its applications,

*Proc. Natl. Acad. Sci. USA***102(23)**(2005), 8109–8113 (electronic). - [22]
V. Rödl and M. Schacht: Regular partitions of hypergraphs: Counting lemmas;

*Combin. Probab. Comput*.**16(6)**(2007), 887–901. - [23]
V. Rödl and M. Schacht: Regular partitions of hypergraphs: Regularity lemmas;

*Combin. Probab. Comput*.**16(6)**(2007), 833–885. - [24]
V. Rödl, M. Schacht, E. Tengan and N. Tokushige: Density theorems and extremal hypergraph problems,

*Israel J. Math*.**152**(2006), 371–380. - [25]
V. Rödl and J. Skokan: Regularity lemma for

*k*-uniform hypergraphs,*Random Structures Algorithms***25(1)**(2004), 1–42. - [26]
V. Rödl and J. Skokan: Applications of the regularity lemma for uniform hypergraphs,

*Random Structures Algorithms***28(2)**(2006), 180–194. - [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. - [28]
J. Solymosi: A note on a question of Erdős and Graham,

*Combin. Probab. Comput*.**13(2)**(2004), 263–267. - [29]
J. Solymosi: Regularity, uniformity, and quasirandomness;

*Proc. Natl. Acad. Sci. USA***102(23)**(2005), 8075–8076 (electronic). - [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. - [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. - [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. - [33]
T. Tao: A variant of the hypergraph removal lemma,

*J. Combin. Theory Ser. A***113(7)**(2006), 1257–1280.

## Author information

### Affiliations

### Corresponding author

## Additional information

A preliminary version of this paper appeared in the proceedings of STOC’ 07.

The first author was partially supported by NSF grants DMS 0300529 and 0800070.

The second author was supported by DFG grant SCHA 1263/1-1.

## Rights and permissions

## About this article

### Cite this article

Rödl, V., Schacht, M. Generalizations of the removal lemma.
*Combinatorica* **29, **467–501 (2009). https://doi.org/10.1007/s00493-009-2320-x

Received:

Published:

Issue Date:

### Mathematics Subject Classification (2000)

- 05C35
- 05C65
- 68W20