Constructing large k-systems on surfaces

Let S g denote the genus g closed orientable surface. For k ∈ 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 S g whose size is on the order of g k / 4 . For each k ≥ 2 , we construct a k-system on S g with on the order of g ⌊ ( 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.