Andreae's ubiquity conjecture for locally finite connected graphs

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Andreae's ubiquity conjecture for locally finite connected graphs

   Given a graph $G$ and a graph relation $\vartriangleleft$, call $G$ $\vartriangleleft$-ubiquitous if, whenever a graph $\Gamma$ satisfies $nG\vartriangleleft\Gamma$ for every $n\in\mathbb{N}$, it also satisfies $\aleph_0G\vartriangleleft\Gamma$, where $\alpha G$ is the disjoint union of $\alpha$ copies of $G$. The minor relation is denoted by $\preccurlyeq$. Andreae's connected-graph conjecture. Every locally finite connected graph is $\preccurlyeq$-ubiquitous. This is the connected-graph form of Andreae's broader ubiquity conjecture and remains open; the paper studies extensive tree-decompositions as a route toward such results.

   source: Nathan Bowler, Christian Elbracht, Joshua Erde, J. Pascal Gollin, Karl Heuer, Max Pitz and Maximilian Teegen, “Ubiquity in graphs III: Ubiquity of locally finite graphs with extensive tree-decompositions”, arXiv:2012.13070 (2021).