  An $\ell$-lift of a graph $G$ is any graph obtained by replacing every vertex
of $G$ with an independent set of size $\ell$, and connecting every pair of two
such independent sets that correspond to an edge in $G$ by a matching of size
$\ell$. Graph lifts have found numerous interesting applications and
connections to a variety of areas over the years. Of particular importance is
the random graph model obtained by considering an $\ell$-lift of a graph
sampled uniformly at random. This model was first introduced by Amit and Linial
in 1999, and has been extensively investigated since. In this paper, we study
the size of the largest topological clique in random lifts of complete graphs.
  In 2006, Drier and Linial raised the conjecture that almost all $\ell$-lifts
of the complete graph on $n$ vertices contain a subdivision of a clique of
order $\Omega(n)$ as a subgraph provided $\ell$ is at least linear in $n$. We
confirm their conjecture in a strong form by showing that for $\ell \ge
(1+o(1))n$, one can almost surely find a subdivision of a clique of order $n$.
We prove that this is tight by showing that for $\ell \le (1-o(1))n$, almost
all $\ell$-lifts do not contain subdivisions of cliques of order $n$.
  Finally, for $2 \le \ell \ll n$, we show that almost all $\ell$-lifts of
$K_n$ contain a subdivision of a clique on $(1-o(1))\sqrt{\frac{2n
\ell}{1-1/\ell}}$ vertices and that this is tight up to the lower order term.
