Colouring geometric overlap graphs

Recently, Pawlik et al. [3] have shown that triangle-free intersection graphs of line segments can have arbitrary large chromatic number, thus answering a long standing conjecture of Erdős [1] in the negative. They show this by constructing a sequence of graphs that have chromatic number tending to infinity. In fact, the chromatic number is O(log logn) where n is the number of vertices. Graphs in this sequence can also all be obtained from a set of axis-parallel rectangles as follows. Take a set F of axis-parallel (hollow) rectangular frames in the plane so that for any two frames that intersect, the right side of one rectangle intersects the top and bottom of the other rectangle (see Figure 1). Then we can construct a graph G where there is a vertex of G labelled by each frame of F and there is an edge between two vertices if the corresponding frames intersect. If G is the set of all graphs that can be obtained this way then a recent result of Krawczyk et al. [2] show that a complementary upper bound of O(log logn) on the chromatic number of all graphs in G. This shows that a “better” sequence of graphs with faster asymptotic growth in the chromatic number is not possible for any construction whose elements remain in G.