On plane spanning trees and cycles of multicolored point sets with few intersections

Let P₁,…,Pk be a collection of disjoint point sets in R² in general position. We prove that for each 1?i?k we can find a plane spanning tree Ti of Pi such that the edges of T₁,…,Tk intersect at most , where n is the number of points in P₁∪?∪Pk. If the intersection of the convex hulls of P₁,…,Pk is nonempty, we can find k spanning cycles such that their edges intersect at most (k−1)n times, this bound is tight. We also prove that if P and Q are disjoint point sets in general position, then the minimum weight spanning trees of P and Q intersect at most 8n times, where |P∪Q|=n (the weight of an edge is its length).