{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T09:33:42Z","timestamp":1746524022632,"version":"3.40.4"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2024,12,23]],"date-time":"2024-12-23T00:00:00Z","timestamp":1734912000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Structural convergence is a framework for the convergence of graphs by Ne\u0161et\u0159il and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline1.png\"\/><jats:tex-math>\n$(G_n)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> converging to a limit <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline2.png\"\/><jats:tex-math>\n$L$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and a vertex <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline3.png\"\/><jats:tex-math>\n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline4.png\"\/><jats:tex-math>\n$L$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, it is possible to find a sequence of vertices <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline5.png\"\/><jats:tex-math>\n$(r_n)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline6.png\"\/><jats:tex-math>\n$L$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> rooted at <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline7.png\"\/><jats:tex-math>\n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the limit of the graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline8.png\"\/><jats:tex-math>\n$G_n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> rooted at <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline9.png\"\/><jats:tex-math>\n$r_n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. A counterexample was found by Christofides and Kr\u00e1l\u2019, but they showed that the statement holds for almost all vertices <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline10.png\"\/><jats:tex-math>\n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline11.png\"\/><jats:tex-math>\n$L$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We offer another perspective on the original problem by considering the size of definable sets to which the root <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline12.png\"\/><jats:tex-math>\n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> belongs. We prove that if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline13.png\"\/><jats:tex-math>\n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000427_inline14.png\"\/><jats:tex-math>\n$(r_n)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> always exists.<\/jats:p>","DOI":"10.1017\/s0963548324000427","type":"journal-article","created":{"date-parts":[[2024,12,23]],"date-time":"2024-12-23T04:04:10Z","timestamp":1734926650000},"page":"392-400","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Structural convergence and algebraic roots"],"prefix":"10.1017","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3566-8214","authenticated-orcid":false,"given":"David","family":"Hartman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-7202-8681","authenticated-orcid":false,"given":"Tom\u00e1\u0161","family":"Hons","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5133-5586","authenticated-orcid":false,"given":"Jaroslav","family":"Ne\u0161et\u0159il","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2024,12,23]]},"reference":[{"key":"S0963548324000427_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0963548324000427_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548315000048"},{"key":"S0963548324000427_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511609589"},{"key":"S0963548324000427_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-33700-8_18"},{"key":"S0963548324000427_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-27875-4"},{"key":"S0963548324000427_ref1","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v6-96"},{"key":"S0963548324000427_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2018.32"},{"key":"S0963548324000427_ref15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139015417"},{"key":"S0963548324000427_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-007-2214-8"},{"key":"S0963548324000427_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2006.05.002"},{"key":"S0963548324000427_ref10","volume-title":"\u201cA Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth\u201d","volume":"263","author":"Ne\u0161et\u0159il","year":"2020"},{"key":"S0963548324000427_ref8","doi-asserted-by":"publisher","DOI":"10.1090\/coll\/060"},{"key":"S0963548324000427_ref6","doi-asserted-by":"publisher","DOI":"10.5817\/CZ.MUNI.EUROCOMB23-075"},{"key":"S0963548324000427_ref16","first-page":"399","volume-title":"Complex Analytic Varieties","author":"Whitney","year":"1972"},{"key":"S0963548324000427_ref5","unstructured":"[5] Hartman, D. , Hons, T. and Ne\u0161et\u0159il, J. (2022) Gadget construction and structural convergence. https:\/\/arxiv.org\/abs\/2212.10985"},{"key":"S0963548324000427_ref11","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20719"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548324000427","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T05:00:45Z","timestamp":1745902845000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548324000427\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,23]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["S0963548324000427"],"URL":"https:\/\/doi.org\/10.1017\/s0963548324000427","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2024,12,23]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}