Abstract: In graph theory, Kneser graph, K(n,k) is the graph whose vertices are k-element subsets of an n-element set and two vertices are adjacent if they are disjoint. In 1955, M. Kneser conjectured that the chromatic number of K(n,k) is n-2k+2. In 1978, Lovász gave a proof of it for the first time, using the connectedness of the neighborhood complex of a graph. Soon after,  Imre Bárány gave a simpler proof using the Borsuk-Ulam theorem and Gale's lemma. In 2002, Joshua Greene further simplified the proof. In this talk, we will discuss the history and the proof given by Joshua Greene.

Abstract: In this talk, I will introduce Borsuk graphs, (circular) chromatic numbers and anti--Vietoris-Rips complexes. I will discuss the homotopy types of the anti--VR complexes and metric thickennings built on S^n upto certain scales. Using topological contradictions, I will show the absence of graph homomorphisms from the Borsuk graph on S^n to the Borsuk graph on S^1 at a certain scale. This gives us a new proof of the lower bound n+2 of the chromatic numbers of Borsuk graphs on S^n. This is joint work with Henry Adams and Alex Elchesen.

Abstract:  In this talk, I will introduce Borsuk graphs, (circular) chromatic numbers, and anti-Vietoris-Rips thickenings. I will discuss the homotopy types of the anti-VR thickenings built on S^n in a range of scales. Using topological obstructions, I will show that for k>n, a graph homomorphisms from the Borsuk graph on S^k to the Borsuk graph on S^n can exist only if the scale for the latter Borsuk graph is sufficiently relaxed. Can similar techniques be used to provide new lower bounds on the chromatic number of Borsuk graphs? This is joint work with Henry Adams, Alex Elchesen and, Michael Moy.