Normal Approximation and Fourth Moment Theorems for Monochromatic Triangles
Abstract
Given a graph sequence \Gn\n ≥ 1 denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with c ≥ 2 colors. This arises as a generalization of the birthday paradox, where Gn corresponds to a friendship network and T3(Gn) counts the number of triples of friends with matching birthdays. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates. The proof involves constructing a martingale difference sequence by carefully ordering the vertices of Gn, based on a certain combinatorial score function, and using a quantitive 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 c ≥ 5. We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any c ≥ 2, which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T3(Gn), whenever c ≥ 5. 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 Bhattacharya et al. (2017).