A Brief Glimpse of Topological Combinatorics

Lovazs’ proof of the Kneser Conjecture presents a beautiful application of Borsuk-Ulam Theorem, a purely topological result, to a combinatorial problem on finite graphs. Moreover, Kneser-Lovasz Theorem is far from being the only case in which a combinatorial problem admits as a solution that uses topological techniques. It is rather surprising that the study of continuous maps would yield elegant solutions to entirely discrete combinatorial problems. This survey seeks to give a brief glimpse of such beautiful topic of topological combinatorics. We survey three kinds of combinatorial problems. The first problem concerns the game hex, and we discuss how one can apply Brouwer’s Fixed Point Theorem to show that the game always has a winner. The second problem is the necklace problem, in which the ham-sandwich theorem provides a nice solution. Lastly, we give a short description of Lovasz’s work on neighborhood complexes, which generalizes the Kneser conjecture and in fact gives a lower bound for the chromatic number of any graph.