A strengthening of Halin's grid theorem

TL;DR

This paper strengthens Halin's grid theorem by demonstrating that for any infinite collection of disjoint equivalent rays in a graph, a subdivision of the hexagonal half-grid can be found with all vertical rays in that collection, offering more control and a shorter proof.

Contribution

It provides a more controlled version of Halin's grid theorem with a simplified proof, specifying the set of rays used in the grid subdivision.

Findings

Existence of a hexagonal half-grid subdivision with prescribed rays

Shorter proof of the strengthened theorem

Enhanced control over rays in grid structures

Abstract

We show that for every infinite collection $\mathcal{R}$ of disjoint equivalent rays in a graph $G$ there is a subdivision of the hexagonal half-grid in $G$ such that all its vertical rays belong to $\mathcal{R}$. This result strengthens Halin's grid theorem by giving control over which specific set of rays is used, while its proof is significantly shorter.