The threshold for loose Hamilton cycles in random hypergraph

TL;DR

We determine the threshold for the appearance of a loose Hamilton cycle in a random $r$-uniform hypergraph $H_{n,m}$ (with $r\ge3$ and $(r-1)\mid n$) as $m$ crosses $\frac{(1+\varepsilon)n\log n}{r}$, matching the known lower bound up to constants. The core method couples $H_{n,m}$ to a multi-type edge process with three edge types and analyses through a Kahn-style framework for perfect matchings, enabling the embedding of $\rho$ independent copies of a $d^*$-out hypergraph to assemble a loose Hamilton cycle. The authors also extend the Shamir problem by incorporating a minimum-degree condition, construct uniform independent perfect matchings in related sub-hypergraphs, and address divisibility via Ferber’s technique plus Friedgut’s no-coarse-threshold argument. Together, these results establish the asymptotically optimal threshold for loose Hamiltonicity and provide a versatile coupling toolkit for random hypergraphs with divisibility and degree constraints.

Abstract

We show that w.h.p.\ the random $r$-uniform hypergraph $H_{n,m}$ contains a loose Hamilton cycle, provided $r\geq 3$ and $m\geq \frac{(1+ε)n\log n}{r}$, where $ε$ is an arbitrary positive constant. This is asymptotically best possible, as if $m\leq \frac{(1-ε)n\log n}{r}$ then w.h.p.\ $H_{n,m}$ contains isolated vertices.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Theorem 7