Generalizations of the removal lemma

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.

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Correspondence to Vojtěch Rödl.

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.

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

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C65
  • 68W20