关于Vera T的问题关于graphons的大小强迫
On a question of Vera T. Sós about size forcing of graphons
摘要：
来自图子$W:[0,1]^2\to [0,1]$的$k$ -样本$\mathbb{G}(k,W)$是$\{1,\dots,k\}$上的随机图，其中我们随机均匀地对$x_1,\dots,x_k\in [0,1]$进行采样，并使每对$\{i,j\}\subseteq \{1,\dots,k\}$成为概率为$W(x_i,x_j)$的边，所有这些选择都是相互独立的。令随机变量$X_k(W)$为$\mathbb{G}(k,W)$中的边数。薇拉·T Sós在2012年提出，如果随机变量$X_k(U)$和$X_k(W)$对每个整数$k\ge 2$都有相同的分布，那么两个图子$U,W$是否必然弱同构。当其中一个图子$W$是一个常数函数时，这个问题由Endre Csóka肯定地回答，并由Jacob Fox，Tomasz Rumuczak和Vera T独立地回答。好的在这里，我们调查的问题时，$W$是一个2步的graphon，并证明了答案是积极的3维家庭这样的graphons。我们也提出了一些相关的结果。
英文摘要：
The $k$-sample $\mathbb{G}(k,W)$ from a graphon $W:[0,1]^2\to [0,1]$ is the random graph on $\{1,\dots,k\}$, where we sample $x_1,\dots,x_k\in [0,1]$ uniformly at random and make each pair $\{i,j\}\subseteq \{1,\dots,k\}$ 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 $\mathbb{G}(k,W)$. 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 $k\ge 2$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
