Finitely forcible graph limits are universal

TL;DR

This paper proves that any graphon can be embedded into a finitely forcible graphon, showing that finitely forcible graphons can have arbitrarily complex structures, contrary to previous conjectures about their simplicity.

Contribution

It demonstrates that all graphons can be realized as subgraphons of finitely forcible graphons, challenging prior beliefs about their structural simplicity.

Findings

Any graphon is a subgraphon of a finitely forcible graphon

Finitely forcible graphons can have arbitrarily complex structures

Minimization problems in extremal graph theory can have complex unique solutions

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.