Positive codegree thresholds for Hamilton cycles in hypergraphs

TL;DR

This paper determines the optimal positive codegree thresholds for Hamilton cycles in hypergraphs, confirming a conjecture for the case of tight Hamilton cycles and revealing a duality with existing codegree conditions.

Contribution

It provides the asymptotically best possible minimum positive codegree conditions for Hamilton -cycles in hypergraphs, including a proof of a recent conjecture.

Findings

Established asymptotically optimal codegree thresholds.

Revealed a duality with codegree conditions.

Confirmed a recent conjecture for tight Hamilton cycles.

Abstract

For each $k \geq 3$ and $1 \leq \ell \leq k-1$ we give an asymptotically best possible minimum positive codegree condition for the existence of a Hamilton $\ell$-cycle in a $k$-uniform hypergraph. This result exhibits an interesting duality with its analogue under a minimum codegree condition. The special case $\ell = k-1$ of our result establishes an asymptotic version of a recent conjecture of Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia on tight Hamilton cycles in hypergraphs.