Limits of dense graph sequences

We show that if a sequence of dense graphs G n has the property that for every fixed graph F, the density of copies of F in G n tends to a limit, then there is a natural “limit object,” namely a symmetric measurable function W : [ 0 , 1 ] 2 → [ 0 , 1 ] . This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. We also characterize graph parameters that are obtained as limits of subgraph densities by the “reflection positivity” property. the way we introduce a rather general model of random graphs, which seems to be interesting on its own right.