We show that posets of bounded height whose cover graphs exclude a fixed
graph as a topological minor have bounded dimension. This result was already
proven by Walczak. However, our argument is entirely combinatorial and does not
rely on structural decomposition theorems. Given a poset with large dimension
but bounded height, we directly find a large clique subdivision in its cover
graph. Therefore, our proof is accessible to readers not familiar with
topological graph theory, and it allows us to provide explicit upper bounds on
the dimension. With the introduced tools we show a second result that is
supporting a conjectured generalization of the previous result. We prove that
(k+k)-free posets whose cover graphs exclude a fixed graph as a topological
minor contain only standard examples of size bounded in terms of k.Comment: revised versio
