A Borsuk–Ulam Equivalent that Directly Implies Sperner's Lemma

Abstract

We show that Fan's 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk––Ulam equivalents, we show that this lemma directly implies Sperner's Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer fixed-point theorem, or that the Lusternik–Schnirelmann–Borsuk theorem implies the KKM lemma.

Additional information

Notes on contributors

Kathryn L. Nyman

KATHRYN L. NYMAN received her B.A. from Carthage College in 1995 and her Ph.D. from Cornell University in 2001. She is an associate professor of mathematics at Willamette University, nestled near waterfalls, wineries, and the occasional alpaca farm in Oregon's Willamette Valley. Her research is in algebraic and enumerative combinatorics. Swing dancing, hiking, painting, and board games keep Kathryn out of trouble. She heartily recommends working with a Los Angeles collaborator to all of her rainy Pacific Northwest and aspiring-extra colleagues; on a research trip for this paper she filmed a webisode of The Guild.

Francis Edward Su

FRANCIS EDWARD SU credits Michael Starbird at UT-Austin for stimulating his interest in fixed-point theorems as an udergraduate. He earned his Ph.D. at Harvard and is now professor of mathematics at Harvey Mudd College. His research is in geometric combinatorics and applications to the social sciences. His passion for popularizing mathematics is evident in his hugely popular Math Fun Facts website and iPhone app, and in his real analysis course on YouTube. In his spare time, gardening and songwriting give him outlets for pondering deep theological questions. Oregonian collaborators can provide opportunities for stardom as well; his appearance as an extra in the movie Blue Like Jazz was filmed in Portland during the writing of this paper, and it may yield him one of the lowest Erdős-Bacon numbers.