EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10408956" target="_blank" >RIV/00216208:11320/19:10408956 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=Z.gR8L6OzS" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=Z.gR8L6OzS</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1017/jsl.2018.32" target="_blank" >10.1017/jsl.2018.32</a>

Result language
angličtina
Original language name
EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES
Original language description
A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed: 1. If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit. 2. A monotone class of graphs C has the property that every FO-convergent sequence of graphs from C has a modeling limit if and only if C is nowhere dense, that is if and only if for each integer p there is N(p) such that no graph in C contains the pth subdivision of a complete graph on N(p) vertices as a subgraph. In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense-somewhere dense dichotomy.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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