Let S be a bicolored set of n points on the plane. A subset I ⊆ S is called an island of S, if I is the intersection of S and a convex set C. In this paper we give an O(n 3 )-time algorithm to find a monochromatic island of maximum cardinality. Our approach also optimizes other parameters and gives an approximation to the class cover problem.

a b s t r a c tLet S be a finite set of geometric objects partitioned into classes or colors. A subset S ′ ⊆ S is said to be balanced if S ′ contains the same amount of elements of S from each of the colors. We study several problems on partitioning 3-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m lines of each color, there is a segment intercepting m lines of each color.(b) Given n red points, n blue points and n green points on any closed Jordan curve γ , we show that for every integer k with 0 ≤ k ≤ n there is a pair of disjoint intervals on γ whose union contains exactly k points of each color. (c) Given a set S of n red points, n blue points and n green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of S.

The pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex. If for each vertex its incident edges have different color, then this maximum is known as achromatic index. Both indices have been widely studied. A geometric graph is a graph drawn in the plane such that its vertices are points in general position, and its edges are straight-line segments. In this paper we extend the notion of pseudoachromatic and achromatic indices for geometric graphs, and present results for complete geometric graphs. In particular, we show that for n points in convex position the achromatic index and the pseudoachromatic index of the complete geometric graph are n 2 +n 4 .

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