Combinatorial Integer Labeling Theorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations - Gerard van der Laan; Dolf Talman; Zaifu Yang (Tinbergen Institute, 2007)
Tucker's well-known combinatorial lemma states that for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set 1,2,...n,-1,-2,....-n with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set 1,2,...n,-1,-2,....-n. Using a constructive approach we prove two combinatorial theorems of Tucker type, stating that under some mild conditions there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the same unit cube. These theorems will be used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions.
Keywords: C61; C62; C68; C72; C58; 330; Sperner lemma; Tucker lemma; integer labeling; simplicial algorithm; discrete nonlinear equations; Nichtlineare Optimierung; Nichtkooperatives Spiel; Theorie