Forcing generalised quasirandom graphs efficiently

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F24%3A00138601" target="_blank" >RIV/00216224:14330/24:00138601 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1017/S0963548323000263" target="_blank" >https://doi.org/10.1017/S0963548323000263</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1017/S0963548323000263" target="_blank" >10.1017/S0963548323000263</a>

Result language
angličtina
Original language name
Forcing generalised quasirandom graphs efficiently
Original language description
We study generalised quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovasz and Sos showed that the structure of such graphs is forced by homomorphism densities of graphs with at most $(10q)<^>q+q$ vertices; subsequently, Lovasz refined the argument to show that graphs with $4(2q+3)<^>8$ vertices suffice. Our results imply that the structure of generalised quasirandom graphs with $qge 2$ parts is forced by homomorphism densities of graphs with at most $4q<^>2-q$ vertices, and, if vertices in distinct parts have distinct degrees, then $2q+1$ vertices suffice. The latter improves the bound of $8q-4$ due to Spencer.
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
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OECD FORD branch
10100 - Mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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