Coarse counterparts of brambles, tangles, and grid minors

Determine whether brambles, tangles, or grid minors admit suitable coarse counterparts that are equivalent to coarse treewidth, and establish the coarse analogue of the Grid Minor Theorem.

Background

The paper places coarse tree decompositions and coarse treewidth within a broader programme seeking coarse analogues of classical structural graph-theoretic tools. In classical graph theory, several parameters and theorems—including bramble number, tangle number, and the Grid Minor Theorem—are closely tied to treewidth. The authors explicitly state that it is unknown whether these notions have appropriate coarse formulations, and that the coarse Grid Minor Theorem remains conjectural.