Colorings of graphs with geometric representations
(Bartosz Walczak)

It is well known that a graph can have arbitrarily large chromatic number without even containing a triangle. By contrast, many natural classes of graphs have the property that the chromatic number is bounded by some function of the clique number (the maximum size of a clique). How large the chromatic number can be in terms of the clique number for graphs arising from geometric representations (interval graphs, rectangle graphs, string graphs, etc.)? This area of research has recently seen a lot of progress, although many problems still remain open. After a brief introduction and overview of the known upper and lower bounds, we will present several techniques for obtaining such bounds, with a particular focus on a recent technique based on connections with on-line graph coloring problems.