Remarks on partitions into expanders

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.

arXiv comment7 pages. Lemma 3.1 has been corrected. Various minor changes throughout