(Communicated by Krzysztof Burdzy) Abstract. Gábor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every ǫ> 0 there is some finite k such that each graph G in the collection can be broken into connected components of size at most k by removing a set of edges, of size at most ǫ |V (G)|. We presently extend this notion to a certain compactification of finite bounded-degree graphs and show that if a sequence of finite graphs converges to a hyperfinite limit, then the sequence itself is hyperfinite. 1.
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Oded Schramm (2008) studied this question.