The minimum vertex degree for an almost-spanning tight cycle in a $3$-uniform hypergraph

TL;DR

This paper proves that a 3-uniform hypergraph with a minimum vertex degree slightly above 5/9 of the maximum possible guarantees an almost-spanning tight cycle, using advanced regularity methods.

Contribution

It establishes the asymptotically optimal minimum degree condition for the existence of an almost-spanning tight cycle in 3-uniform hypergraphs.

Findings

Minimum degree threshold of (5/9 + o(1)) * binom(n, 2) for almost-spanning tight cycles

Use of hypergraph regularity lemma in the proof

Bound is asymptotically best possible

Abstract

We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, B\"ottcher, Cooley and Mycroft.