Constructing large k-systems on Surfaces

Let $S_{g}$ denote the genus $g$ closed orientable surface. For $k\in \mathbb{N}$, a $k$-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than $k$ times. Juvan-Malni\v{c}-Mohar \cite{Ju-Mal-Mo} showed that there exists a $k$-system on $S_{g}$ whose size is on the order of $g^{k/4}$. For each $k\geq 2$, We construct a $k$-system on $S_{g}$ with on the order of $g^{\lfloor (k+1)/2 \rfloor +1}$ elements. The $k$-systems we construct behave well with respect to subsurface inclusion, analogously to how a pants decomposition contains pants decompositions of lower complexity subsurfaces.