  K\'{a}rolyi, Pach, and T\'{o}th proved that every 2-edge-colored
straight-line drawing of the complete graph contains a monochromatic plane
spanning tree. It is open if this statement generalizes to other classes of
drawings, specifically, to simple drawings of the complete graph. These are
drawings where edges are represented by Jordan arcs, any two of which intersect
at most once. We present two partial results towards such a generalization.
First, we show that the statement holds for cylindrical simple drawings. (In a
cylindrical drawing, all vertices are placed on two concentric circles and no
edge crosses either circle.) Second, we introduce a relaxation of the problem
in which the graph is $k$-edge-colored, and the target structure must be
hypochromatic, that is, avoid (at least) one color class. In this setting, we
show that every $\lceil (n+5)/6\rceil$-edge-colored monotone simple drawing of
$K_n$ contains a hypochromatic plane spanning tree. (In a monotone drawing,
every edge is represented as an $x$-monotone curve.)
