Forcing large tight components in 3-graphs

TL;DR

This paper investigates the minimum codegree conditions in 3-graphs that guarantee the existence of large spanning tight components, providing bounds that approach each other asymptotically.

Contribution

The authors establish bounds on codegree thresholds that force large tight components in 3-graphs, advancing understanding of extremal conditions for spanning structures.

Findings

Minimum codegree at least n/3 guarantees a spanning tight component.

Below this threshold, no tight component can span more than 2n/3 vertices.

Bounds on codegree thresholds for various tight component sizes are asymptotically tight.

Abstract

Any $n$-vertex $3$-graph with minimum codegree at least $\lfloor n/3\rfloor$ must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than $\lceil 2n/3\rceil$ vertices. Motivated by this observation, we ask which codegree forces a tight component of at least any given size. The corresponding function seems to have infinitely many discontinuities, but we provide upper and lower bounds, which asymptotically converge as the function nears the origin.