4.
Given a graph sequence denote by
T3(
Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of
Gn with colors. In this paper we prove a central limit theorem (CLT) for
T3(
Gn) 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(
Gn), 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].
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