Counterexample to the conjectured coarse grid theorem

TL;DR

This paper constructs a counterexample to the conjectured coarse grid theorem, showing certain graphs defy expected quasi-isometric and minor properties, thereby refuting longstanding conjectures in graph theory.

Contribution

It provides a counterexample to the coarse grid theorem and related conjectures, demonstrating limitations of existing graph minor and quasi-isometry theories.

Findings

Counterexample to the coarse grid theorem

Planar graphs without the coarse Erdős-Pósa property

Modification of recent counterexamples to related conjectures

Abstract

We show that for every $M,A,n \in \mathbb{N}$ there exists a graph $G$ that does not contain the $(154\times 154)$-grid as a $3$-fat minor and is not $(M,A)$-quasi-isometric to a graph with no $K_n$ minor. This refutes the conjectured coarse grid theorem by Georgakopoulos and Papasoglu and the weak fat minor conjecture of Davies, Hickingbotham, Illingworth, and McCarty. Our construction is a slight modification of the recent counterexample to the weak coarse Menger conjecture from Nguyen, Scott and Seymour. We further modify the construction to show that there are planar graphs that do not have the coarse Erd\H{o}s-P\'{o}sa property.