19 problems

Fat minor conjecture. For every graph $X$ there exists a function $f:\mathbb{N}\to\mathbb{N}^2$ such that, for every graph $G$ and $K\in\mathbb{N}$, if $G$ does not contain $X$ as…

Coarse Menger conjecture. There is a function $g:\mathbb{N}^2\to\mathbb{N}$ such that for every $k,d\in\mathbb{N}$, every graph or geodesic metric space $G$, and every two sets…

Let $G$ be a finite or infinite graph and let $A\subseteq V(G)$. An $A$-path is a path in $G$ between two distinct vertices of $A$. Two subgraphs are at distance at least $r$ when…

Coarse Erdős–Pósa conjecture. There exist functions

Bounded-cycle-space fat minor conjecture. For every graph $X$ there exists a function $f:\mathbb{N}^2\to\mathbb{N}^2$ such that, for every graph $G$ whose cycle space is generated…

Planar graph coarse Menger conjecture. Every planar graph has the coarse Menger property.

Let $G$ be a graph and let $X,Y\subseteq V(G)$. An $X$-$Y$ path is a path with at least one end in $X$ and at least one end in $Y$. A set is $(a,b)$-centered if it is contained in…

Let $\Sigma$ be a fixed surface. Let $G$ be a graph embeddable in $\Sigma$, and let $X,Y\subseteq V(G)$. An $X$-$Y$ path is a path with at least one end in $X$ and at least one end…

Let $G$ be a finite or infinite graph, and let $X,Y\subseteq V(G)$. An $X$-$Y$ path is a path with at least one end in $X$ and at least one end in $Y$. A set $Z\subseteq V(G)$ is…

Let $G$ be a finite or infinite graph, and let $X,Y\subseteq V(G)$. An $X$-$Y$ path is a path with at least one end in $X$ and at least one end in $Y$. For sets $S,T\subseteq V(G)$…

Let $J$ be a finite graph and let $K\in\mathbb N$. A graph has no $K$-fat $J$ minor if it contains no geometric $J$ minor whose branch sets and connecting paths are mutually at dis…

Let $K,n,m,r\in\mathbb{N}$. A graph is $(M,A)$-quasi-isometric to a graph of tree-width at most $g$ in the usual coarse sense, and connected sets are at least $K$ apart when every…

Let $K,d\in\mathbb{N}$, and let $H$ be a graph. A graph forbids a $K$-fat $H$ minor if it contains no such fat minor model. Then there exist some $M,A\in\mathbb{N}$ and a graph…

Let $K,d,n\in\mathbb{N}$. A graph has a $K$-fat $(n\times n)$-grid minor when it contains the corresponding fat minor model. Then there exists some $g\in\mathbb{N}$ such that every…

Let $n\in\mathbb{N}$. An induced $(n\times n)$-grid minor is the indicated induced-minor model, and $(M,A)$-quasi-isometry and tree-width have their usual meanings. Then there exis…

Let $K,n \in \mathbb{N}$. A graph has a $K$-fat $(n\times n)$-grid minor if it contains the corresponding fat minor model, and graphs are $(M,A)$-quasi-isometric when they satisfy…

Georgakopoulos's coarse grid minor conjecture. For every planar graph $H$, there exist $k,q\in\mathbb{N}$ such that every $H$-induced-minor-free graph is $q$-quasi-isometric to a g…

Nguyen–Scott–Seymour conjecture. There is a constant $k$ such that if a graph $G$ admits a quasi-isometry to a graph of tree-width at most two, then $G$ admits a quasi-isometry wit…

Relaxed coarse Menger conjecture. For all $k,d$ there exist $\ell,n$ such that, if $S,T$ are sets of vertices in a graph $G$, then either …