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.

中文翻译：

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分区扩展器的说明

在本注释中，我们提供了一个简短的证明，即可以将不具有线性小Følner集的图划分为一个扩展器的并集。我们利用这一事实证明了允许线性小极大Følner集的图的划分结果，并推论出这样的图族必须包含一个扩展器族。我们还表明，扩展器中分区的存在是准等距不变的。