Abstract

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 ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ 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.