EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed: 1. If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit. 2. A monotone class of graphs C has the property that every FO-convergent sequence of graphs from C has a modeling limit if and only if C is nowhere dense, that is if and only if for each integer p there is N(p) such that no graph in C contains the pth subdivision of a complete graph on N(p) vertices as a subgraph. In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense-somewhere dense dichotomy.