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On a question of Vera T. Sos about size forcing of graphons
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Cooley, Oliver, Kang, Mihyun and Pikhurko, Oleg (2022) On a question of Vera T. Sos about size forcing of graphons. Acta Mathematica Hungarica, 168 . pp. 1-26. doi:10.1007/s10474-022-01265-8 ISSN 0236-5294.
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Official URL: https://doi.org/10.1007/s10474-022-01265-8
Abstract
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.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Random graphs, Graph theory, Forcing (Model theory), Combinatorial analysis, Linear operators, Discrete mathematics | ||||||||||||
Journal or Publication Title: | Acta Mathematica Hungarica | ||||||||||||
Publisher: | Springer ; Akademiai Kiado Rt | ||||||||||||
ISSN: | 0236-5294 | ||||||||||||
Official Date: | October 2022 | ||||||||||||
Dates: |
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Volume: | 168 | ||||||||||||
Page Range: | pp. 1-26 | ||||||||||||
DOI: | 10.1007/s10474-022-01265-8 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Re-use Statement: | This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s10474-022-01265-8 | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 22 July 2022 | ||||||||||||
Date of first compliant Open Access: | 23 November 2023 | ||||||||||||
RIOXX Funder/Project Grant: |
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