We bound the error for the normal approximation of the number of triangles in the Erdős–Rényi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein bounds obtained by Barbour et al. [(1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B 47: 125–145], resolving a long-standing open problem. The proofs are based on a new variant of the Stein–Tikhomirov method—a combination of Stein's method and characteristic functions introduced by Tikhomirov [(1976). The rate of convergence in the central limit theorem for weakly dependent variables. Vestnik Leningradskogo Universiteta 158–159, 166].

我们限制了 Erdős–Rényi 随机图中三角形数量关于 Kolmogorov 度量的正态逼近的误差。我们的界限与 Barbour等人获得的最佳可用 Wasserstein 界限相匹配。[（1989）。可分解随机变量的中心极限定理，适用于随机图。Journal of Combinatorial Theory, Series B 47: 125–145]，解决了一个长期存在的开放性问题。证明是基于 Stein-Tikhomirov 方法的一个新变体——结合了 Stein 方法和 Tikhomirov [(1976) 引入的特征函数。弱因变量中心极限定理的收敛速度。Vestnik Leningradskogo Universiteta 158–159, 166]。