  A topological graph drawn on a cylinder whose base is horizontal is
\emph{angularly monotone} if every vertical line intersects every edge at most
once. Let $c(n)$ denote the maximum number $c$ such that every simple angularly
monotone drawing of a complete graph on $n$ vertices contains at least $c$
pairwise disjoint edges. We show that for every simple complete topological
graph $G$ there exists $\Delta$, $0<\Delta<n$, such that $G$ contains at least
$\max \{\frac n\Delta, c(\Delta)\}$ pairwise disjoint edges. By combining our
result with a result of T\'oth we obtain an alternative proof for the best
known lower bound of $\Omega(n^\frac 13)$ on the maximum number of pairwise
disjoint edges in a simple complete topological graph proved by Suk. Our proof
is based on a result of Ruiz-Vargas.
