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

The k-sample from a graphon W: [ 0, 1 ]²→ [ 0, 1 ] is the random graph on { 1, ⋯, k}, where we sample x₁, ⋯, x_k∈ [ 0, 1 ] uniformly at random and make each pair { i, j} ⊆ { 1, ⋯, k} an edge with probability W(x_i, x_j), with all these choices being mutually independent. Let the random variable X_k(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 X_k(U) and X_k(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.