On a Question of Vera T. Sós About Size Forcing of Graphons

Oliver Cooley*, Mihyun Kang, Oleg Pikhurko

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

The k-sample 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. Vera T. Sós asked in 2012 whether two graphons U, W are necessarily weakly isomorphic provided 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.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Pages625-630
Number of pages6
DOIs
Publication statusPublished - 2021

Publication series

NameTrends in Mathematics
Volume14
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Keywords

  • Graphons
  • Sample
  • Weak isomorphism

ASJC Scopus subject areas

  • Mathematics(all)

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