TY - CHAP

T1 - On a Question of Vera T. Sós About Size Forcing of Graphons

AU - Cooley, Oliver

AU - Kang, Mihyun

AU - Pikhurko, Oleg

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Graphons

KW - Sample

KW - Weak isomorphism

UR - http://www.scopus.com/inward/record.url?scp=85114106342&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-83823-2_100

DO - 10.1007/978-3-030-83823-2_100

M3 - Chapter

AN - SCOPUS:85114106342

T3 - Trends in Mathematics

SP - 625

EP - 630

BT - Trends in Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -