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]

A colouring of a graph is nonrepetitive if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive colourings with a bounded number of colours, thus proving a conjecture of Alon, Grytczuk, Hałuszczak and Riordan (2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding a fixed minor, and graphs excluding a fixed topological minor.

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.

An outerstring graph is an intersection graph of curves that lie in a common halfplane and have one endpoint on the boundary of that half-plane. We prove that the class of outerstring graphs is χ-bounded, which means that their chromatic number is bounded by a function of their clique number. This generalizes a series of previous results on χ-boundedness of outerstring graphs with various additional restrictions on the shape of curves or the number of times the pairs of curves can cross. The assumption that each curve has an endpoint on the boundary of the half-plane is justified by the known fact that triangle-free intersection graphs of straight-line segments can have arbitrarily large chromatic number.A preliminary version of this paper appeared in1 Outerstring realizations corresponding to the two definitions can be mapped to each other by a homeomorphism transforming a closed half-plane to a closed disc without one boundary point. 2 Specifically, it is proved in [45, Corollary 2.7] that a graph is a Hasse diagram if and only if it is triangle-free and its complement is a cylinder graph. 3 This is mentioned in [31] in Note added in proof and follows from the above and the fact that triangle-free complements of outerstring graphs are Hasse diagrams, which is a direct consequence of Theorem 1 in [59]. 4 Given a graph G, add a vertex r non-adjacent to G and, for every vertex v of G, a vertex rv adjacent only to r and v. The graph thus obtained is a string graph if and only if G is an outerstring graph; see [5, Lemma 1]. 5 One needs the fact that the graph G4 defined in [52] is not an outerstring graph. This is because any outerstring representation of G4 could be extended to a string representation of G5, contradicting Lemma 3.2 in [52].