Abstract
In this note we give a short proof that graphs having no linearly small Følner sets can be partitioned into a union of expanders. We use this fact to prove a partition result for graphs admitting linearly small maximal Følner sets and we deduce that a family of such graphs must contain a family of expanders. We also show that the existence of partitions into expanders is a quasi-isometry invariant.
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Notes
This is not an important difference: it is not hard to modify our proof to cover this case as well.
This includes some normalization terms that take into account the degrees of vertices. It is natural to consider the conductance when one is planning to use the spectral characterization of expansion (as Oveis Gharam–Trevisan do). In this note we preferred the approach via Cheeger constants because it is marginally simpler to introduce.
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Acknowledgements
I am very grateful to the anonymous referee for pointing out a mistake in the proof of Lemma 3.1. I would like to thank Emmanuel Breuillard for directing me to the paper [3] and pointing out the argument explained in Remark 3.3. I am also thankful to Henry Bradford, Ana Khukhro, Kang Li and Jiawen Zhang for their helpful comments. This work was supported by the ISF Moked 713510 Grant Number 2919/19.
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