**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.