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

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, … , 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 G(k, W). Vera T. Sós asked in 2012 whether two graphons U, W are necessarily weakly isomorphic if 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.