On a question of Vera T. Sós about size forcing of graphons

O. Cooley*, M. Kang, Oleg Pikhurko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The k-sampleG(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.

Original languageEnglish
Pages (from-to)1-26
Number of pages26
JournalActa Mathematica Hungarica
Issue number1
Publication statusPublished - Oct 2022


  • graph container
  • graphon
  • graphon forcing
  • k-sample

ASJC Scopus subject areas

  • Mathematics(all)


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