Session B2 - Graph Theory and Combinatorics
July 13, 17:00 ~ 17:50 - Room B7
Patrice Ossona de Mendez
Centre d’analyse et de mathématique sociales (CAMS) and CNRS, France - firstname.lastname@example.org
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 sequences 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:
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 $\mathcal C$ 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 the $p$th subdivision of the complete graph on $N(p)$ vertices does not belong to $\mathcal C$.
In this talk we present the proof of 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.