In combinatorics and discrete geometry there are many colorful and rainbow results. Additionally topological tools have proven to be useful in generating interesting combinatorial results. In this line of approach we prove a colorful generalization of the Borsuk–Ulam theorem. We give a short proof of Ky Fan’s Lemma and generalize it from involutions to larger symmetry groups Z/p for prime p. We also present colorful generalizations of these nonexistence results for equivariant maps. As consequences we derive a colorful generalization of the Ham–Sandwich theorem and the necklace splitting theorem among other applications of Borsuk-Ulam. This is joint work with Florian Frick.
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