A Sperner labeling of the regular triangulation of the integer simplex Δ_{k,q} has between binomial(q+k-3,k-2) and q^(k-1)-(q-1)^(k-1) non-monochromatic cells.
Homotopies and transcendental extensions in colouring problems
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We develop the technique of geometric realizations with algebraically independent (over the field of real algebraic numbers) coordinates of vertices and combine it with the oriented volume method inspired by work of McLennan and Tourky on the Sperner's lemma. This enables us to prove new results: the non-draw property of the generalized Y game, the theorem about triangulation of the product of two simplices, multilabeled Ky Fan' s lemma, and give new proofs of known results: the multilabeled version of Sperner's lemma and generalized Atanassov conjecture.
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On the minimum number of non-monochromatic simplices for Sperner labelings of a regular triangulation
A Sperner labeling of the regular triangulation of the integer simplex Δ_{k,q} has between binomial(q+k-3,k-2) and q^(k-1)-(q-1)^(k-1) non-monochromatic cells.