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 $g^{k\mathrm{/}4}$. For each $k\ge 2$, We construct a $k$-system on $S_g$ with on the order of $g^{\lfloor (k+1)/2 \rfloor +1}$elements\mathrm{.}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.