Three Edge-disjoint Plane Spanning Paths in a Point Set

We consider the following problem: Given a set $S$ of $n$ distinct points in the plane, how many edge-disjoint plane straight-line spanning paths can be drawn on $S$? Each spanning path must be crossing-free, but edges from different paths are allowed to intersect at arbitrary points. It is known that if the points of $S$ are in convex position, then $\lfloor n/2 \rfloor$ such paths always exist. However, for general point sets, the best known construction yields only two edge-disjoint plane spanning paths. In this paper, we prove that for any set $S$ of at least ten points in general position (i.e., no three points are collinear), it is always possible to draw at least three edge-disjoint plane straight-line spanning paths. Our proof relies on a structural result about halving lines in point sets and builds on the known two-path construction, which we also strengthen: we show that for any set $S$ of at least six points, and for any two specified points on the boundary of the convex hull of $S$, there exist two edge-disjoint plane spanning paths that start at those prescribed points. Finally, we complement our positive results with a lower bound: for every $n \geq 6$, there exists a set of $n$ points for which no more than $\lceil n/3 \rceil$ edge-disjoint plane spanning paths are possible.