The Symmetry Preserving Removal Lemma

In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemerédi’s Theorem on arithmetic progressions. If in an Abelian group A there are sets S1, S2 . . . , St such that the number of arithmetic progressions x1, x2, . . . , xt with xi ∈ Si is o(|A| ) then we can shrink each Si by o(|A|) elements such that the new sets don’t have such a diagonal arithmetic progression.