EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2019

Kód důvěrnosti údajů

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

Název periodika

Journal of Symbolic Logic

ISSN

0022-4812

e-ISSN

—

Svazek periodika

84

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

21

Strana od-do

452-472

Kód UT WoS článku

000470903600002

EID výsledku v databázi Scopus

2-s2.0-85064599120