Small filling sets of curves on a surface

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 ⩾ 1 times is 2 g / K as g → ∞ . 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.