The symmetry preserving removal lemma

In this paper we observe that in the hypergraph removal lemma, the edge removal can be done in such a way that the symmetries of the original hypergraph remain preserved. As an application we prove the following generalization of Szemeredi's Theorem on arithmetic progressions. Let A be an Abelian group with subsets S 1 , S 2 ,...,S t such that the number of arithmetic progressions x, x + d,..., x + (t - 1)d with x + (i - 1)d ∈ S i is o(|A| 2 ). Then we can shrink each S i by o(|A|) elements such that the new sets don't have any arithmetic progression of the above type.