Loose Hamilton cycles in hypergraphs

TL;DR

This paper proves that dense enough k-uniform hypergraphs contain loose Hamilton cycles, extending previous results and simplifying the proof using the hypergraph blow-up lemma.

Contribution

It establishes a minimum degree condition for the existence of loose Hamilton cycles in k-uniform hypergraphs, generalizing prior work and employing a simplified proof technique.

Findings

Minimum degree threshold for loose Hamilton cycles in hypergraphs

Application of the hypergraph blow-up lemma to simplify proofs

Extension of known results from 3-uniform to k-uniform hypergraphs

Abstract

We prove that any k-uniform hypergraph on n vertices with minimum degree at least n/(2(k-1))+o(n) contains a loose Hamilton cycle. The proof strategy is similar to that used by K\"uhn and Osthus for the 3-uniform case. Though some additional difficulties arise in the k-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash.