The Symmetry Preserving Removal Lemma

TL;DR

This paper introduces a symmetry-preserving version of the hyper-graph removal lemma and applies it to generalize Szemerédi's theorem, showing how to eliminate certain arithmetic progressions while maintaining structural symmetries.

Contribution

It presents a symmetry-preserving approach to the hyper-graph removal lemma and extends Szemerédi's theorem to sets with sparse arithmetic progressions in Abelian groups.

Findings

Symmetry-preserving edge removal in hyper-graphs.

Generalization of Szemerédi's theorem for Abelian groups.

Ability to eliminate sparse arithmetic progressions while maintaining symmetry.

Abstract

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.