On Borsuk’s conjecture

In my lecture I will present a classical conjecture of Borsuk on partitioning of sets in metric spaces into parts of smaller diameter. I will discuss both well-established results and new developments: for example, the refutation of Borsuk’s conjecture on spheres of small radius and on existence of diameter graphs with a large girth (without cycles of a length smaller than a certain predetermined value). A number of unsolved problems will be presented, many of which can be solved by a first-year student.