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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].

KOLMOGOROV BOUNDS FOR THE NORMAL APPROXIMATION OF THE NUMBER OF TRIANGLES IN THE ERDŐS–RÉNYI RANDOM GRAPH