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@article{1524557, author = {Cooper, Jacob and Kráľ, Daniel and Martins, TL}, article_location = {SAN DIEGO}, article_number = {December}, doi = {http://dx.doi.org/10.1016/j.aim.2018.10.019}, keywords = {Graph limits; Extremal graph theory}, language = {eng}, issn = {0001-8708}, journal = {Advances in Mathematics}, title = {Finitely forcible graph limits are universal}, url = {https://arxiv.org/abs/1701.03846}, volume = {340}, year = {2018} }
TY - JOUR ID - 1524557 AU - Cooper, Jacob - Kráľ, Daniel - Martins, TL PY - 2018 TI - Finitely forcible graph limits are universal JF - Advances in Mathematics VL - 340 IS - December SP - 819-854 EP - 819-854 PB - ACADEMIC PRESS INC ELSEVIER SCIENCE SN - 00018708 KW - Graph limits KW - Extremal graph theory UR - https://arxiv.org/abs/1701.03846 L2 - https://arxiv.org/abs/1701.03846 N2 - 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. ER -
COOPER, Jacob, Daniel KRÁĽ and TL MARTINS. Finitely forcible graph limits are universal. \textit{Advances in Mathematics}. SAN DIEGO: ACADEMIC PRESS INC ELSEVIER SCIENCE, 2018, vol.~340, December, p.~819-854. ISSN~0001-8708. Available from: https://dx.doi.org/10.1016/j.aim.2018.10.019.
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