Constructing large k-systems on surfaces

Let SgSg denote the genus g   closed orientable surface. For k∈Nk∈N, a k-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than k times. Juvan–Malnič–Mohar [3] showed that there exists a k  -system on SgSg whose size is on the order of gk/4gk/4. For each k≥2k≥2, we construct a k  -system on SgSg with on the order of g⌊(k+1)/2⌋+1g⌊(k+1)/2⌋+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.