Packing curves on surfaces with few intersections

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 a polynomial in $|\chi(S)|$ of degree $k^{2}+k+1$. In this paper, we narrow Przytycki's bounds by showing that $$\mathcal{N}_{k}(S) =O \left( \frac{ |\chi|^{3k}}{ ( \log |\chi| )^2 } \right) ,$$ In particular, the size of a maximal 1-system grows sub-cubically in $|\chi(S)|$. The proof uses a circle packing argument of Aougab-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 can improve our estimates using a different argument to give $$\mathcal{N}_{k}(S) \leq O(n^{2k+2}) .$$ Using similar techniques, we also obtain the sharp estimate $\mathcal{N}_{2}(S)=\Theta(n^3)$ when $k=2$ and $g$ is fixed.