Title: Finitely forcible graph limits are universal,
Abstract: The theory of graph limits represents large graphs by analytic objects called
graphons. Graph limits determined by finitely many graph densities, which are
represented by finitely forcible graphons, arise in various scenarios,
particularly within extremal combinatorics. Lovasz and Szegedy conjectured that
all such graphons possess a simple structure, e.g., the space of their typical
vertices is always finite dimensional; this was disproved by several ad hoc
constructions of complex finitely forcible graphons. We prove that any graphon
is a subgraphon of a finitely forcible graphon. This dismisses any hope for a
result showing that finitely forcible graphons possess a simple structure, and
is surprising when contrasted with the fact that finitely forcible graphons
form a meager set in the space of all graphons. In addition, since any finitely
forcible graphon represents the unique minimizer of some linear combination of
densities of subgraphs, our result also shows that such minimization problems,
which conceptually are among the simplest kind within extremal graph theory,
may in fact have unique optimal solutions with arbitrarily complex structure.