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The symmetry preserving removal lemma

Author: Balázs Szegedy
Journal: Proc. Amer. Math. Soc. 138 (2010), 405-408
MSC (2000): Primary 05C99
Published electronically: October 14, 2009
MathSciNet review: 2557157
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Abstract: 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.

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Balázs Szegedy
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S-2E4, Canada

Received by editor(s): September 16, 2008
Received by editor(s) in revised form: November 20, 2008
Published electronically: October 14, 2009
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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