  A fundamental result in structural graph theory states that every graph with
large average degree contains a large complete graph as a minor. We prove this
result with the extra property that the minor is small with respect to the
order of the whole graph. More precisely, we describe functions $f$ and $h$
such that every graph with $n$ vertices and average degree at least $f(t)$
contains a $K_t$-model with at most $h(t)\cdot\log n$ vertices. The logarithmic
dependence on $n$ is best possible (for fixed $t$). In general, we prove that
$f(t)\leq 2^{t-1}+\eps$. For $t\leq 4$, we determine the least value of $f(t)$;
in particular $f(3)=2+\eps$ and $f(4)=4+\eps$. For $t\leq4$, we establish
similar results for graphs embedded on surfaces, where the size of the
$K_t$-model is bounded (for fixed $t$).
