{"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","month":"11","title":"On a question of Vera T. Sós about size forcing of graphons","publication":"Acta Mathematica Hungarica","external_id":{"isi":["000886839900006"],"arxiv":["2103.09114"]},"date_created":"2023-01-12T12:07:59Z","oa_version":"Preprint","language":[{"iso":"eng"}],"publication_status":"published","page":"1-26","type":"journal_article","date_published":"2022-11-23T00:00:00Z","publisher":"Springer Nature","abstract":[{"text":"The k-sample G(k,W) from a graphon W:[0,1]2→[0,1] is the random graph on {1,…,k}, where we sample x1,…,xk∈[0,1] uniformly at random and make each pair {i,j}⊆{1,…,k} an edge with probability W(xi,xj), with all these choices being mutually independent. Let the random variable Xk(W) be the number of edges in  G(k,W). Vera T. Sós asked in 2012 whether two graphons U, W are necessarily weakly isomorphic if the random variables Xk(U) and Xk(W) have the same distribution for every integer k≥2. This question when one of the graphons W is a constant function was answered positively by Endre Csóka and independently by Jacob Fox, Tomasz Łuczak and Vera T. Sós. Here we investigate the question when W is a 2-step graphon and prove that the answer is positive for a 3-dimensional family of such graphons. We also present some related results.","lang":"eng"}],"article_processing_charge":"No","intvolume":"       168","status":"public","date_updated":"2023-08-04T09:02:37Z","keyword":["graphon","k-sample","graphon forcing","graph container"],"doi":"10.1007/s10474-022-01265-8","department":[{"_id":"MaKw"}],"quality_controlled":"1","day":"23","acknowledgement":"Supported by Austrian Science Fund (FWF) Grant I3747. Supported by ERC Advanced Grant 101020255 and Leverhulme Research Project Grant RPG-2018-424.\r\nAn extended abstract of this paper appeared in the Proceedings of the European Conference\r\non Combinatorics, Graph Theory and Applications (EuroComb 2021), CRM Research Perspectives, Springer.","citation":{"short":"O. Cooley, M. Kang, O. Pikhurko, Acta Mathematica Hungarica 168 (2022) 1–26.","ama":"Cooley O, Kang M, Pikhurko O. On a question of Vera T. Sós about size forcing of graphons. <i>Acta Mathematica Hungarica</i>. 2022;168:1-26. doi:<a href=\"https://doi.org/10.1007/s10474-022-01265-8\">10.1007/s10474-022-01265-8</a>","ieee":"O. Cooley, M. Kang, and O. Pikhurko, “On a question of Vera T. Sós about size forcing of graphons,” <i>Acta Mathematica Hungarica</i>, vol. 168. Springer Nature, pp. 1–26, 2022.","apa":"Cooley, O., Kang, M., &#38; Pikhurko, O. (2022). On a question of Vera T. Sós about size forcing of graphons. <i>Acta Mathematica Hungarica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10474-022-01265-8\">https://doi.org/10.1007/s10474-022-01265-8</a>","chicago":"Cooley, Oliver, M. Kang, and O. Pikhurko. “On a Question of Vera T. Sós about Size Forcing of Graphons.” <i>Acta Mathematica Hungarica</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s10474-022-01265-8\">https://doi.org/10.1007/s10474-022-01265-8</a>.","mla":"Cooley, Oliver, et al. “On a Question of Vera T. Sós about Size Forcing of Graphons.” <i>Acta Mathematica Hungarica</i>, vol. 168, Springer Nature, 2022, pp. 1–26, doi:<a href=\"https://doi.org/10.1007/s10474-022-01265-8\">10.1007/s10474-022-01265-8</a>.","ista":"Cooley O, Kang M, Pikhurko O. 2022. On a question of Vera T. Sós about size forcing of graphons. Acta Mathematica Hungarica. 168, 1–26."},"oa":1,"volume":168,"main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2103.09114"}],"_id":"12151","year":"2022","scopus_import":"1","article_type":"original","isi":1,"publication_identifier":{"issn":["0236-5294"],"eissn":["1588-2632"]},"author":[{"id":"43f4ddd0-a46b-11ec-8df6-ef3703bd721d","first_name":"Oliver","last_name":"Cooley","full_name":"Cooley, Oliver"},{"first_name":"M.","last_name":"Kang","full_name":"Kang, M."},{"first_name":"O.","last_name":"Pikhurko","full_name":"Pikhurko, O."}]}