Colorful Equivariant Topological Methods in Discrete Mathematics

A map is equivariant if it preserves symmetries. Starting with the classical Borsuk–Ulam theorem, which states that a map between spheres that preserves antipodal symmetry cannot decrease dimension, equivariant topology has developed a plethora of obstructions for the existence of certain symmetry-preserving maps. The resulting methods have proven particularly fruitful in combinatorial problems. This thesis further refines equivariant topological methods with a particular eye towards combinatorial applications. Colorful (or rainbow) results are popular across combinatorics, and we develop colorful generalizations of equivariant topological results. Colorful results have the following general form: Whereas the original result asserts the existence of a k-element subset with a property P of a certain set S, a colorful generalization asserts the existence of a k-transversal with property P of k such sets. We prove a colorful generalization of the Borsuk–Ulam theorem and generalize it to larger groups of symmetries. As applications we derive colorful generalizations of hyperplane equipartition results. Furthermore, we give novel applications of related colorful techniques to zero-sum Ramsey theory. The foundation of zero-sum Ramsey theory is a result of Erdős, Ginzburg, and Ziv which states that given any 2n−1 integers, some n of those integers will sum to zero modulo n. Among others, we generalize this result to an equivariant-topological criterion for the existence of zero-sum hyperedges modulo n in integer-valued colorings of hypergraphs.