  In this paper, we investigate the problem of separating a set $X$ of points
in $\mathbb{R}^{2}$ with an arrangement of $K$ lines such that each cell
contains an asymptotically equal number of points (up to a constant ratio). We
consider a property of curves called the stabbing number, defined to be the
maximum countable number of intersections possible between the curve and a line
in the plane. We show that large subsets of $X$ lying on Jordan curves of low
stabbing number are an obstacle to equal separation. We further discuss Jordan
curves of minimal stabbing number containing $X$. Our results generalize recent
bounds on the Erd\H{o}s-Szekeres Conjecture, showing that for fixed $d$ and
sufficiently large $n$, if $|X| \ge 2^{c_dn/d + o(n)}$ with $c_d = 1 +
O(\frac{1}{\sqrt{d}})$, then there exists a subset of $n$ points lying on a
Jordan curve with stabbing number at most $d$.
