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Finitely forcible graph limits are universal

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COOPER Jacob KRÁĽ Daniel MARTINS TL

Druh Článek v odborném periodiku
Časopis / Zdroj Advances in Mathematics
Fakulta / Pracoviště MU

Fakulta informatiky

Citace
WWW https://arxiv.org/abs/1701.03846
Doi http://dx.doi.org/10.1016/j.aim.2018.10.019
Klíčová slova Graph limits; Extremal graph theory
Popis 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.