Joint Poisson Convergence of Monochromatic Hyperedges in Multiplex Hypergraphs

TL;DR

This paper proves that the joint distribution of monochromatic hyperedges in multiplex hypergraphs converges to a (possibly dependent) Poisson distribution based on the convergence of first two moments, extending previous graph coloring results.

Contribution

It generalizes the second moment phenomenon for Poisson approximation from graphs to hypergraphs and from marginal to joint convergence in multiplex hypergraphs.

Findings

Joint distribution of monochromatic hyperedges converges to Poisson distributions.

Convergence depends only on the first two moments of the distribution.

Applications include generalized birthday problems and counting monochromatic structures.

Abstract

Given a sequence of $r$-uniform hypergraphs $H_n$, denote by $T(H_n)$ the number of monochromatic hyperedges when the vertices of $H_n$ are colored uniformly at random with $c = c_n$ colors. In this paper, we study the joint distribution of monochromatic hyperedges for hypergraphs with multiple layers (multiplex hypergraphs). Specifically, we consider the joint distribution of ${\bf T} _n:= (T(H_n^{(1)}), T(H_n^{(2)}))$, for two sequences of hypergraphs $H_n^{(1)}$ and $H_n^{(2)}$ on the same set of vertices. We will show that the joint distribution of ${\bf T}_n$ converges to (possibly dependent) Poisson distributions whenever the mean vector and the covariance matrix of ${\bf T}_n$ converge. In other words, the joint Poisson approximation of ${\bf T}_n$ is determined only by the convergence of its first two moments. This generalizes recent results on the second moment phenomenon for Poisson approximation from graph coloring to hypergraph coloring and from marginal convergence to joint convergence. Applications include generalizations of the birthday problem, counting monochromatic subgraphs in randomly colored graphs, and counting monochromatic arithmetic progressions in randomly colored integers. Extensions to random hypergraphs and weighted hypergraphs are also discussed.