End-Devouring Rays

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A ray is a one-way infinite simple path, equipped with its first (starting) vertex. Two rays belong to the same end of a graph $G$ if, after removing any finite set of vertices, their tails lie in the same component. An $\omega$-ray represents the end $\omega$. A family of rays devours $\omega$ if every $\omega$-ray intersects at least one family member. An end is countable in the convention of the source if some countable family of rays devours it; the graph itself need not be countable.

Statement recorded in the supplied source.

For every graph $G$, every countable end $\omega$, and every infinite family $K$ of pairwise vertex-disjoint $\omega$-rays, does there exist a family $K'$ of pairwise vertex-disjoint $\omega$-rays which devours $\omega$ and whose set of starting vertices is exactly that of $K$? [S2] [S3]

Short English statement

Can any prescribed disjoint infinite family of rays in a countable end be replaced, retaining exactly its starting vertices, by disjoint rays meeting every ray in that end?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3306 (OPG-735). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, End-Devouring Rays, node 735, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/end_devouring_rays.
• S3 J. Pascal Gollin and Karl Heuer, Infinite end-devouring sets of rays with prescribed start vertices (2017), abstract. https://arxiv.org/abs/1704.06577.