Finitely forcible graph limits are universal

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F18%3A00106842" target="_blank" >RIV/00216224:14330/18:00106842 - isvavai.cz</a>

Výsledek na webu

<a href="https://arxiv.org/abs/1701.03846" target="_blank" >https://arxiv.org/abs/1701.03846</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.aim.2018.10.019" target="_blank" >10.1016/j.aim.2018.10.019</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Finitely forcible graph limits are universal

Popis výsledku v původním jazyce

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. (C) 2018 Elsevier Inc. All rights reserved.

Název v anglickém jazyce

Finitely forcible graph limits are universal

Popis výsledku anglicky

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. (C) 2018 Elsevier Inc. All rights reserved.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

V - Vyzkumna aktivita podporovana z jinych verejnych zdroju

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Advances in Mathematics

ISSN

0001-8708

e-ISSN

—

Svazek periodika

340

Číslo periodika v rámci svazku

December

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

36

Strana od-do

819-854

Kód UT WoS článku

000451363700020

EID výsledku v databázi Scopus

2-s2.0-85055086923