Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].

中文翻译：

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单色三角形的正态近似和四阶矩定理

给定图序列由分别表示Ť ₃（g ^ _Ñ）单色三角形中的顶点的均匀随机着色数量ģ _Ñ与颜色。在本文中，我们使用马丁格尔 CLT 的定量版本证明了具有明确错误率的T ₃ ( G _n )的中心极限定理 (CLT) 。然后我们将这个误差项与众所周知的第四时刻现象联系起来，有趣的是，只有当颜色数量满足 时才成立。我们还表明，四阶矩的收敛对于获得任何高斯极限是必要的。，这与上述结果一起，意味着四_阶矩条件刻画了T ₃ ( G _n )的极限正态分布，无论何时。最后，为了说明我们方法的前景，我们包含了单色边缘数量的 CLT 的替代证明，它为 [7] 中获得的结果提供了定量率。