---
_id: '12151'
abstract:
- lang: eng
  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.
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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Oliver
  full_name: Cooley, Oliver
  id: 43f4ddd0-a46b-11ec-8df6-ef3703bd721d
  last_name: Cooley
- first_name: M.
  full_name: Kang, M.
  last_name: Kang
- first_name: O.
  full_name: Pikhurko, O.
  last_name: Pikhurko
citation:
  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>
  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>.
  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.
  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.
  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>.
  short: O. Cooley, M. Kang, O. Pikhurko, Acta Mathematica Hungarica 168 (2022) 1–26.
corr_author: '1'
date_created: 2023-01-12T12:07:59Z
date_published: 2022-11-23T00:00:00Z
date_updated: 2024-10-09T21:03:37Z
day: '23'
department:
- _id: MaKw
doi: 10.1007/s10474-022-01265-8
external_id:
  arxiv:
  - '2103.09114'
  isi:
  - '000886839900006'
intvolume: '       168'
isi: 1
keyword:
- graphon
- k-sample
- graphon forcing
- graph container
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2103.09114'
month: '11'
oa: 1
oa_version: Preprint
page: 1-26
publication: Acta Mathematica Hungarica
publication_identifier:
  eissn:
  - 1588-2632
  issn:
  - 0236-5294
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On a question of Vera T. Sós about size forcing of graphons
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 168
year: '2022'
...