A fourth‐moment phenomenon for asymptotic normality of monochromatic subgraphs

Abstract

Given a graph sequence and a simple connected subgraph , we denote by the number of monochromatic copies of in a uniformly random vertex coloring of with colors. We prove a central limit theorem for (we denote the appropriately centered and rescaled statistic as ) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of which we call good joins. Good joins are closely related to the fourth moment of , which allows us to show a fourth moment phenomenon for the central limit theorem. For , we show that converges in distribution to whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when .