On the decomposition of random hypergraphs

TL;DR

This paper investigates the minimum partition of edges into complete r-partite subhypergraphs in random hypergraphs, establishing probabilistic bounds related to Turán densities for typical hypergraphs.

Contribution

It provides the first probabilistic analysis of the hypergraph partition number for random hypergraphs, linking it to Turán densities and extending classical bipartite results.

Findings

With high probability, f(H) approximates (1 - π(K^{(r-1)}_r)) times the binomial coefficient for certain p.

The result generalizes bipartite hypergraph partitioning to r-uniform hypergraphs.

The paper establishes asymptotic formulas for the hypergraph decomposition number in random models.

Abstract

For an $r$-uniform hypergraph $H$, let $f(H)$ be the minimum number of complete $r$-partite $r$-uniform subhypergraphs of $H$ whose edge sets partition the edge set of $H$. For a graph $G$, $f(G)$ is the bipartition number of $G$ which was introduced by Graham and Pollak in 1971. In 1988, Erd\H{o}s conjectured that if $G \in G(n,1/2)$, then with high probability $f(G)=n-\alpha(G)$, where $\alpha(G)$ is the independence number of $G$. This conjecture and related problems have received a lot of attention recently. In this paper, we study the value of $f(H)$ for a typical $r$-uniform hypergraph $H$. More precisely, we prove that if $(\log n)^{2.001}/n \leq p \leq 1/2$ and $H \in H^{(r)}(n,p)$, then with high probability $f(H)=(1-\pi(K^{(r-1)}_r)+o(1))\binom{n}{r-1}$, where $\pi(K^{(r-1)}_r)$ is the Tur\'an density of $K^{(r-1)}_r$.