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 -