  We show that the asymptotic growth rate for the minimal cardinality of a set
of simple closed curves on a closed surface of genus $g$ which fill and
pairwise intersect at most $K\ge 1$ times is $2\sqrt{g}/\sqrt{K}$ as $g \to
\infty$ . We then bound from below the cardinality of a filling set of systoles
by $g/\log(g)$. This illustrates that the topological condition that a set of
curves pairwise intersect at most once is quite far from the geometric
condition that such a set of curves can arise as systoles.
