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Przytycki has shown that the size   $\mathcal{N}_{k}(S)$   of a maximal collection of simple closed curves that pairwise intersect at most   $k$   times on a topological surface   $S$   grows at most as   $|\chi(S)|^{k^{2}+k+1}$  . In this article, we narrow Przytycki’s bounds, obtaining    \[ \mathcal{N}_{k}(S) =O \left( \frac{ |\chi|^{3k}}{ ( \log |\chi| )^2 } \right)\!. \]    In particular, the size of a maximal 1-system grows sub-cubically in   $|\chi(S)|$  . The proof uses a circle packing argument of Aougab and Souto and a bound for the number of curves of length at most   $L$   on a hyperbolic surface. When the genus   $g$   is fixed and the number of punctures   $n$   grows, we use a different argument to show    \[ \mathcal{N}_{k}(S) \leq O(n^{2k+2}). \]    This may be improved when   $k=2$  , and we obtain the sharp estimate   $\mathcal{N}_2(S)=\Theta(n^3)$  .

Packing Curves on Surfaces with Few Intersections