article published yes Oliver Cooley author 43f4ddd0-a46b-11ec-8df6-ef3703bd721d M. Kang author O. Pikhurko author MaKw department The k-sample G(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. Springer Nature2022 eng graphonk-samplegraphon forcinggraph container 0236-5294 1588-2632 2103.09114 00088683990000610.1007/s10474-022-01265-8 1681-26 Cooley O, Kang M, Pikhurko O. On a question of Vera T. Sós about size forcing of graphons. <i>Acta Mathematica Hungarica</i>. 2022;168:1-26. doi:<a href="https://doi.org/10.1007/s10474-022-01265-8">10.1007/s10474-022-01265-8</a> Cooley, Oliver, M. Kang, and O. Pikhurko. “On a Question of Vera T. Sós about Size Forcing of Graphons.” <i>Acta Mathematica Hungarica</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s10474-022-01265-8">https://doi.org/10.1007/s10474-022-01265-8</a>. O. Cooley, M. Kang, and O. Pikhurko, “On a question of Vera T. Sós about size forcing of graphons,” <i>Acta Mathematica Hungarica</i>, vol. 168. Springer Nature, pp. 1–26, 2022. O. Cooley, M. Kang, O. Pikhurko, Acta Mathematica Hungarica 168 (2022) 1–26. Cooley O, Kang M, Pikhurko O. 2022. On a question of Vera T. Sós about size forcing of graphons. Acta Mathematica Hungarica. 168, 1–26. Cooley, Oliver, et al. “On a Question of Vera T. Sós about Size Forcing of Graphons.” <i>Acta Mathematica Hungarica</i>, vol. 168, Springer Nature, 2022, pp. 1–26, doi:<a href="https://doi.org/10.1007/s10474-022-01265-8">10.1007/s10474-022-01265-8</a>. Cooley, O., Kang, M., &#38; Pikhurko, O. (2022). On a question of Vera T. Sós about size forcing of graphons. <i>Acta Mathematica Hungarica</i>. Springer Nature. <a href="https://doi.org/10.1007/s10474-022-01265-8">https://doi.org/10.1007/s10474-022-01265-8</a> 121512023-01-12T12:07:59Z2024-10-09T21:03:37Z