  We present several results in extremal graph and hypergraph theory of
topological nature. First, we show that if $\alpha>0$ and
$\ell=\Omega(\frac{1}{\alpha}\log\frac{1}{\alpha})$ is an odd integer, then
every graph $G$ with $n$ vertices and at least $n^{1+\alpha}$ edges contains an
$\ell$-subdivision of the complete graph $K_t$, where $t=n^{\Theta(\alpha)}$.
Also, this remains true if in addition the edges of $G$ are properly colored,
and one wants to find a rainbow copy of such a subdivision. In the sparser
regime, we show that properly edge colored graphs on $n$ vertices with average
degree $(\log n)^{2+o(1)}$ contain rainbow cycles, while average degree $(\log
n)^{6+o(1)}$ guarantees rainbow subdivisions of $K_t$ for any fixed $t$, thus
improving recent results of Janzer and Jiang et al., respectively. Furthermore,
we consider certain topological notions of cycles in pure simplicial complexes
(uniform hypergraphs). We show that if $G$ is a $2$-dimensional pure simplicial
complex ($3$-graph) with $n$ $1$-dimensional and at least $n^{1+\alpha}$
2-dimensional faces, then $G$ contains a triangulation of the cylinder and the
M\"obius strip with $O(\frac{1}{\alpha}\log\frac{1}{\alpha})$ vertices. We
present generalizations of this for higher dimensional pure simplicial
complexes as well. In order to prove these results, we consider certain
(properly edge colored) graphs and hypergraphs $G$ with strong expansion. We
argue that if one randomly samples the vertices (and colors) of $G$ with not
too small probability, then many pairs of vertices are connected by a short
path whose vertices (and colors) are from the sampled set, with high
probability.
