Existence of Modeling Limits for Sequences of Sparse Structures

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, and it was conjectured that this is the case if the graphs in the sequence are sufficiently sparse. Precisely, two conjectures were proposed: * 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. * A monotone class of graphs $\mathcal{C}$ has the property that every FO-convergent sequence of graphs from $\mathcal{C}$ has a modeling limit if and only if $\mathcal{C}$ is nowhere dense, that is if and only if for each integer $p$ there is $N(p)$ such that no graph in $\mathcal{C}$ contains the $p$th subdivision of a complete graph on $N(p)$ vertices as a subgraph.

There was a flaw in the proof. Another approach had to be used, and allowed to extend the result to monadically stable classes of structures (see arXiv:2508.08960 [math.LO])