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

Generalizations of the removal lemma

  • Vojtěch Rödl
  • Mathias Schacht


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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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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