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We investigate which classes of infinite graphs have the Erd\H{o}s-P\'osa property (EPP). In addition to the usual EPP, we also consider the following infinite variant of the EPP: a class $\mathcal{G}$ of graphs has the $\kappa$-EPP, where $\kappa$ is an infinite cardinal, if for any graph $\Gamma$ there are either $\kappa$ disjoint graphs from $\mathcal{G}$ in $\Gamma$ or there is a set $X$ of vertices of $\Gamma$ of size less than $\kappa$ such that $\Gamma - X$ contains no graph from $\mathcal{G}$. In particular, we study the ($\kappa$-)EPP for classes consisting of a single infinite graph $G$. We obtain positive results when the set of induced subgraphs of $G$ is labelled well-quasi-ordered, and negative results when $G$ is not a proper subgraph of itself (both results require some additional conditions). As a corollary, we obtain that every graph which does not contain a path of length $n$ for some $n \in \mathbb{N}$ has the EPP and the $\kappa$-EPP. Furthermore, we show that the class of all subdivisions of any tree $T$ has the $\kappa$-EPP for every uncountable cardinal $\kappa$, and if $T$ is rayless, also the $\aleph_0$-EPP and the EPP., Comment: 25 pages, 5 figures

A digraph $H$ is called ubiquitous if every digraph that contains arbitrarily many vertex-disjoint copies of $H$ also contains infinitely many vertex-disjoint copies of $H$. We study oriented double rays, that is, digraphs $H$ whose underlying undirected graphs are double rays. Calling a vertex of an oriented double ray a turn if it has in-degree or out-degree 2, we prove that an oriented double ray with at least one turn is ubiquitous if and only if it has a (finite) odd number of turns. It remains an open problem to determine whether the consistently oriented double ray is ubiquitous., Comment: 11 pages, 1 figure, minor changes

Let $W$ be any wheel graph and $\mathcal{G}$ the class of all countable graphs not containing $W$ as a minor. We show that there exists a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as an induced subgraph., Comment: 14 pages, 2 figures

Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly $\aleph_1$ distinct topological types of countable trees. In general, for any infinite cardinal $\kappa$ there are exactly $\kappa^+$ distinct topological types of trees of size $\kappa$. This solves a problem of van der Holst from 2005., Comment: 7 pages, 1 figure

Call a digraph $H$ \emph{ubiquitous} if every digraph $D$ that contains $k$ vertex-disjoint copies of $H$ for every $k \in \mathbb{N}$ also contains infinitely many vertex-disjoint copies of $H$. We characterise which digraphs whose underlying undirected graph is a ray are ubiquitous., Comment: 12 pages, 3 figures

We present a systematic investigation into how tree-decompositions of finite adhesion capture topological properties of the space formed by a graph together with its ends. As main results, we characterise when the ends of a graph can be distinguished, and characterise which subsets of ends can be displayed by a tree-decomposition of finite adhesion. In particular, we show that a subset $\Psi$ of the ends of a graph $G$ can be displayed by a tree-decomposition of finite adhesion if and only if $\Psi$ is $G_\delta$ (a countable intersection of open sets) in $|G|$, the topological space formed by a graph together with its ends. Since the undominated ends of a graph are easily seen to be $G_\delta$, this provides a structural explanation for Carmesin's result that the set of undominated ends can always be displayed., Comment: 28 pages, 6 figures. Small changes based on referee comments. Accepted for publication in Journal of Combinatorial Theory, Series B

A subgraph-universal graph/a topological minor-universal graph in a class of graphs $\mathcal{G}$ is a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as a subgraph/topological minor. We prove that the class $\mathcal{P}$ of all countable planar graphs does not contain a topological minor-universal graph. This answers a question of Diestel and K\"uhn and strengthens a result of Pach stating that there is no subgraph-universal graph in $\mathcal{P}$. Furthermore, we characterise for which subdivided stars $T$ there is a topological minor-universal graph in the class of all countable $T$-free graphs., Comment: 15 pages, 3 figures. Changes based on referee comments. To appear in Journal of Graph Theory