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 Szemerédi's Theorem on arithmetic progressions. Let $A$ be an Abelian group with subsets $S_1,S_2,\dots,S_t$ such that the number of arithmetic progressions $x,x+d,\dots,x+(t-1)d$ with $x+(i-1)d\in S_i$ is $o(\vert A\vert^2)$. Then we can shrink each $S_i$ by $o(\vert A\vert)$ elements such that the new sets don't have any arithmetic progression of the above type.