Compactness and finite forcibility of graphons

Proc. London Math. Soc.

2018

Self Cite

Graphons are analytic objects associated with convergent sequences of dense graphs. Finitely forcible graphons, that is, those determined by finitely many subgraph densities, are of particular interest because of their relation to various problems in extremal combinatorics and theoretical computer science. Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon always has finite dimension, which would have implications on the minimum number of parts in its weak ε‐regular partition. We disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices that has infinite dimension.

“…A graphon

W

satisfies a constraint

D_{1} = D_{2}

, if both

D_{1}

and

D_{2}

are equal when evaluated with each

H

substituted with

d (H, W)

. As it was observed in , if a graphon

W

is a unique (up to weak isomorphism) graphon that satisfies a finite set

scriptC

of constraints, then the graphon

W

is finitely forcible. In particular,

W

is the unique (up to weak isomorphism) graphon with densities of graphs appearing in

scriptC

equal to their densities in

W

.…”

Section: Definitionsmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 90%

“…Following the framework from , when proving that a graphon is finitely forcible, we give a set of constraints that uniquely determines

W

rather than listing the finitely many graphs and their densities that uniquely determine

W

. A constraint is an equality between two density expressions, where a density expression is a formal real polynomial combination of graphs, that is, a real number or a graph

H

are density expressions, and if

D_{1}

and

D_{2}

are two density expressions, then the sum

D_{1} + D_{2}

and the product

D_{1} \cdot D_{2}

are also density expressions.…”

Section: Definitionsmentioning

confidence: 99%

“…In his construction, both

T (W)

and

\bar{T} false(W false)

contain a subspace homeomorphic to

{false[0, 1 false]}^{d}

. 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

{false[0, 1 false]}^{\infty}

Section: Introductionmentioning

confidence: 99%

“…We finish with giving a brief outline of the proof of Theorem in informal terms. As in , the constructed hypercubical graphon

W_{⊞}

has several parts (see Figure ), which are determined by the degrees of the vertices that are contained in the parts. The parts

A_{1}, \dots, A_{3}

serve to further partition the parts

B_{1}, \dots, B_{5}

into infinitely many smaller parts; the part

B_{1}

is split into parts

B_{1, d}

d \in N

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Infinite‐dimensional finitely forcible graphon

Glebov

Klimošová

Proc. London Math. Soc.

2018

Self Cite

Finitely forcible graph limits are universal

Cooper

Advances in Mathematics

Martins

2018

Self Cite

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.

Elusive extremal graphs

Grzesik

Proc. London Math. Soc.

Lovász

2020

We study the uniqueness of optimal solutions to extremal graph theory problems. Lovász conjectured that every finite feasible set of subgraph density constraints can be extended further by a finite set of density constraints so that the resulting set is satisfied by an asymptotically unique graph. This statement is often referred to as saying that 'every extremal graph theory problem has a finitely forcible optimum'. We present a counterexample to the conjecture. Our techniques also extend to a more general setting involving other types of constraints.