We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological fixed-point theorems. Our work naturally suggests several interesting open questions for future research.
CITATION STYLE
Grant, E., & Ma, W. (2013). A geometric approach to combinatorial fixed-point theorems: extended abstract. In The Seventh European Conference on Combinatorics, Graph Theory and Applications (pp. 463–468). Scuola Normale Superiore. https://doi.org/10.1007/978-88-7642-475-5_74
Mendeley helps you to discover research relevant for your work.