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.