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. Lovász 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.

Infinite‐dimensional finitely forcible graphon

“…We now analyze the constraints depicted in Figure 20. We now show that g ∞ satisfies (13). The third constraint in Figure 20 implies that…”

“…for almost every b ∈ B 1,k . Since the image of g k is dense in [0, 1] k even after removing a set of measure zero from its domain, we conclude that g ∞ satisfies (13). It follows that ψ D is measure preserving.…”

“…The line of research on constructions of complex finitely forcible graph limits culminated with the result of Cooper et al . that every graphon is a subgraphon of a finitely forcible graphon. This general result of Cooper et al .…”

“…It is worth noting that while for any ε > 0, there is a graphon W such that each weak ε-regular W has at least 2 Ω(ε −2 ) parts, there is no graphon such that every weak ε-regular partition of W is at least 2 Ω(ε −2 ) parts for an infinite sequence of ε tending to zero. The line of research on constructions of complex finitely forcible graph limits culminated with the result of Cooper et al [13] that every graphon is a subgraphon of a finitely forcible graphon. This general result of Cooper et al was also a key ingredient in the argument of Grzesik et al in [20] for disproving a conjecture of Lovász that every extremal graph theory problem has a finitely forcible optimum, which was one of the most cited open problems on dense graph limits.…”

“…One of the contributions of this paper is showing how the techniques from [] and Norine [Private Communication] can be refined to force a subspace homeomorphic to , which turned out to be quite challenging. Another contribution of the paper is formalizing the methods used in [] and Norine [Private Communication], which are further used in the follow up papers .…”