On the Intersections of Non-homotopic Loops

Abstrakt

Let V={v1,…,vn} be a set of n points in the plane and let x∈V . An x-loop is a continuous closed curve not containing any point of V, except of passing exactly once through the point x. We say that two x-loops are non-homotopic if they cannot be transformed continuously into each other without passing through a point of V. For n=2 , we give an upper bound 2O(k) on the maximum size of a family of pairwise non-homotopic x-loops such that every loop has fewer than k self-intersections and any two loops have fewer than k intersections. This result is inspired by a very recent result of Pach, Tardos, and Tóth who proved the upper bounds 216k4 for the slightly different scenario when x∉V .