A combinatorial analog of a theorem of F.J. Dyson

Tuckerʹs 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 PhD 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 Dysonʹs theorem. As with Tuckerʹs lemma, we hope that this will lead to generalizations and applications and ultimately a combinatorial analog of Yangʹs theorem of which both Borsuk–Ulam and Dyson are special cases.