13 problems

Linear Erdős–Pósa conjecture. At least one of the following holds:

Coarse Erdős–Pósa conjecture. There exist functions

For a digraph $D$, a set of directed cycles has distinct lengths when no two of its cycles have the same length. Half-integral distinct-length directed-cycle conjecture. For every…

For a graph $G$, let $\Lambda(G)=\{\ell\in\mathbb{N}\mid G\text{ contains a cycle of length }\ell\}$, and let $G-X$ denote the graph obtained by deleting the vertices in $X$. Disti…

Let $G$ be a graph and let $A$ be a subset of the vertices of $G$. An $A$-path is a path in $G$ whose two endpoints belong to $A$. For subsets $X$ of vertices of $G$ and integers…

Coarse Erdős--Pósa theorem. There are $f$ and $C$ as above satisfying this assertion. Alternatively, when $X$ is a graph, removing $B_{CK}(S)$ results in a quasi-forest.

For an abelian group $\Gamma$, a subset $A\subseteq\Gamma$, and a $\Gamma$-labelled graph $(G,\gamma)$, let $\mathcal O^A_{G,\gamma}$ denote the set of cycles whose $\gamma$-values…

Uniform wheel-model Erdős–Pósa conjecture. There are a constant $c>0$ and a function $g:\mathbb{N}\to\mathbb{N}$ such that, for every integer $t\geq 3$, wheel models in $G$ have th…

Treewidth packing conjecture. There is a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G$ with

The planar-minor Erdős–Pósa conjecture. There is a constant depending on $H$ such that

Let $l\geq 6$ be an integer, and let $\mathcal{C}_{\geq l}$ denote the class of cycles of length at least $l$. For a graph class $\mathcal{H}$, let…

Let $G$ be a planar graph and let $u{ell}(G)$ and $tau{ell}(G)$ denote, respectively, the packing number and transversality of its hypergraph of balls of radius $ell$. **Chepoi–Es…

Let $g \colon \mathbb{N}\rightarrow\mathbb{N}$ be a function such that every graph excluding an $r$-vertex planar graph as a minor has treewidth at most $g(r)$. Robertson–Seymour–T…