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Dedicated to Endre Szemerédi on the occasion of his 70th birthday In this paper we highlight a topological aspect of the graph limit theory. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size.

Regularity Partitions and The Topology of Graphons