The Symmetry Preserving Removal Lemma

Details The Symmetry Preserving Removal Lemma The Symmetry Preserving Removal Lemma

2008-09-15

arxiv.org/abs/0809.2626v1

Mathematics

Combinatorics

Scientific paper

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 Szemer\'edi's Theorem on arithmetic progressions. If in an Abelian group $A$ there are sets $S_1,S_2...,S_t$ such that the number of arithmetic progressions $x_1,x_2,...,x_t$ with $x_i\in S_i$ is $o(|A|^2)$ then we can shrink each $S_i$ by $o(|A|)$ elements such that the new sets don't have such a diagonal arithmetic progression.

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