A Combinatorial Analog of a Theorem of F.J.Dyson

Pallavi Jayawant, Peter Wong

Published 2006-08-08, updated 2009-11-18Version 3

Tucker's Lemma is a combinatorial analog of the Borsuk-Ulam theorem and the case n=2 was proposed by Tucker in 1945. Numerous generalizations and applications of the Lemma have appeared since then. In 2006 Meunier proved the Lemma in its full generality in his Ph.D. thesis. There are generalizations and extensions of the Borsuk-Ulam theorem that do not yet have combinatorial analogs. In this note, we give a combinatorial analog of a result of Freeman J. Dyson and show that our result is equivalent to Dyson's theorem. As with Tucker's Lemma, we hope that this will lead to generalizations and applications and ultimately a combinatorial analog of Yang's theorem of which both Borsuk-Ulam and Dyson are special cases.