Chromatic Number of G-Borsuk Graphs

Explore the chromatic number of G-Borsuk graphs in this 50-minute lecture from the Applied Algebraic Topology Network. Delve into the definition of G-Borsuk graphs, their relationship to compact spaces with free group actions, and the connection between their chromatic number and the topology of the underlying space. Examine lower bounds using G-actions on Hom-complexes and upper bounds derived from a recursive formula on the space's dimension. Investigate the conjecture that the true chromatic number matches the lower bound, supported by computational evidence. Study random G-Borsuk graphs and the thresholds for epsilon that maintain the chromatic number of the whole graph. Analyze the tightness of results when the G-index and dimension of the space coincide, and explore the transition of chromatic numbers in various scenarios.