In the 1970s, Erdős asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer k, we construct a triangle-free family of line segments in the plane with chromatic number greater than k. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number. arXiv:1209.1595v5 [math.CO]

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set X in R 2 that is not an axis-aligned rectangle and for any positive integer k produces a family F of sets, each obtained by an independent horizontal and vertical scaling and translation of X , such that no three sets in F pairwise intersect and χ(F) > k. This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

Recently, it was proved that triangle-free intersection graphs of n line segments in the plane can have chromatic number as large as (log log n). Essentially the same construction produces (log log n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes-those belonging to any class of compact arcconnected sets in R 2 closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number O(log log n), improving on the previous bound of O(log n). To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with O(log log n) colors Preliminary version of this paper appeared as: Coloring triangle-free rectangular frame intersection graphs with O(log log n) colors. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.), Graph-Theoretic Concepts in Computer Science (WG 2013). Lecture Notes in Computer Science, vol. 8165, pp. 333-344. Springer, Berlin (2013 Geom (2015) 53:199-220 using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line L if the intersection of any member with L is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.

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