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ABSTRACT A digraph H $H$ is called ubiquitous if every digraph that contains arbitrarily many vertex‐disjoint copies of H $H$ also contains infinitely many vertex‐disjoint copies of H $H$. We study oriented double rays, that is, digraphs H $H$ 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. [ABSTRACT FROM AUTHOR]

Let W be any wheel graph and G the class of all countable graphs not containing W as a minor. We show that there exists a graph in G which contains every graph in G as an induced subgraph. [ABSTRACT FROM AUTHOR]

A digraph H $H$ is called ubiquitous if every digraph D $D$ that contains k $k$ vertex‐disjoint copies of H $H$ for every k∈N $k\in {\mathbb{N}}$ also contains infinitely many vertex‐disjoint copies of H $H$. We characterise which digraphs with rays as underlying undirected graphs are ubiquitous. [ABSTRACT FROM AUTHOR]

A subgraph‐universal graph/a topological minor‐universal graph in a class of graphs G ${\mathscr{G}}$ is a graph in G ${\mathscr{G}}$ which contains every graph in G ${\mathscr{G}}$ as a subgraph/topological minor. We prove that the class P ${\mathscr{P}}$ of all countable planar graphs does not contain a topological minor‐universal graph. This answers a question of Diestel and Kühn and strengthens a result of Pach stating that there is no subgraph‐universal graph in P ${\mathscr{P}}$. Furthermore, we characterise for which subdivided stars T $T$ there is a topological minor‐universal graph in the class of all countable T $T$‐free graphs. [ABSTRACT FROM AUTHOR]