Topological designs

Benson Farb and Chris Leininger had asked how many pairwise non-isotopic simple closed curves can be placed on a surface of genus \(g\) in such a way that any two of the curves intersect at most once. In this note we use combinatorial methods to give bounds (a lower bound of \((g+1)g\) curves, and an exponential upper bound). While the bounds for the general Farb/Leininger question are (conjecturally) weak, the results presented here are of independent interest.