TY - JOUR

T1 - On a question of Vera T. Sós about size forcing of graphons

AU - Cooley, O.

AU - Kang, M.

AU - Pikhurko, Oleg

N1 - Funding Information:
Supported by Austrian Science Fund (FWF) Grant I3747.
Funding Information:
Supported by ERC Advanced Grant 101020255 and Leverhulme Research Project Grant RPG-2018-424.
Publisher Copyright:
© 2022, Akadémiai Kiadó, Budapest, Hungary.

PY - 2022/10

Y1 - 2022/10

N2 - 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.

AB - 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.

KW - graph container

KW - graphon

KW - graphon forcing

KW - k-sample

UR - http://www.scopus.com/inward/record.url?scp=85142413602&partnerID=8YFLogxK

U2 - 10.1007/s10474-022-01265-8

DO - 10.1007/s10474-022-01265-8

M3 - Article

AN - SCOPUS:85142413602

VL - 168

SP - 1

EP - 26

JO - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1

ER -