First-Order Convergence and Roots

Abstract

Nesetril and Ossona de Mendez introduced the notion of first order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether if $(G_i)_{i\in\mathbb{N}}$ is a sequence of graphs with M being their first order limit and v is a vertex of M, then there exists a sequence $(v_i)_{i\in\mathbb{N}}$ of vertices such that the graphs G_i rooted at v_i converge to M rooted at v. We show that this holds for almost all vertices v of M and we give an example showing that the statement need not hold for all vertices.