Abstract

Let $V = {v_1, \dots, v_n}$ be a set of $n$ points in the plane and let $x \in V$. 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 $2^{O(k)}$ 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 $2^{16k^4}$ for the slightly different scenario when $x\not\in V$.