Red point blue point

Probability Level 5

In a plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are co-linear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the minimal value of $k$ such that the goal is attainable for every possible configuration of 4027 points.