Coarse Erdős–Pósa conjecture for fat triangle minor-models

A graph $G$ contains a $d$-fat minor-model of $K_3$ when the branch sets and connecting paths satisfy the coarse separation conditions associated with fat minors; write $B_G(X,r)$ for the set of vertices of $G$ at distance at most $r$ from $X$. Two such models are pairwise at distance more than $d$ when their vertex sets are mutually at distance greater than $d$.

Coarse Erdős–Pósa conjecture. There exist functions

$$f:\mathbb{N}\to\mathbb{N}\qquad\text{and}\qquad g:\mathbb{N}\to\mathbb{N},$$

with $g(d)\in\mathcal{O}(d)$, such that, for all positive integers $k$ and $d$, and every graph $G$, either $G$ contains $k$ $d$-fat minor-models of $K_3$ that are pairwise at distance more than $d$, or there exists a subset $X$ of vertices of $G$ with $|X|\leqslant f(k)$ such that $G-B_G(X,g(d))$ has no $d$-fat $K_3$ minor-model.

This is a coarse analogue of the Erdős–Pósa theorem, replacing ordinary minor-models by fat minor-models and requiring the models to be far apart. The source reports that a proof was announced in March 2026, but supplies no definitive resolution; the database status therefore remains open pending verification of that announcement.