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Huynh et al. recently showed that a countable graph $G$ which contains everycountable planar graph as a subgraph must contain arbitrarily large finitecomplete graphs as topological minors, and an infinite complete graph as aminor. We strengthen this result by showing that the same conclusion holds, if$G$ contains every countable planar graph as a topological minor. Inparticular, there is no countable planar graph containing every countableplanar graph as a topological minor, answering a question by Diestel andK\"uhn.  Moreover, we construct a locally finite planar graph which contains everylocally finite planar graph as a topological minor. This shows that in theabove result it is not enough to require that $G$ contains every locally finiteplanar graph as a topological minor.

Universal planar graphs for the topological minor relation