Conflict-free coloring for rectangle ranges using O(n ^.382) colors

Given a set of points P ⊆ ℝ ², a conflict-free coloring of P w. r. t. rectangle ranges is an assignment of colors to points of P, such that each nonempty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to base stations (points) such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in ℝ ² can be conflict-free colored with O(n ^β*+o(1)) colors in expected polynomial time, where.