Combinatorial Integer Labeling Thorems on Finite Sets With an Application to Discrete Systems of Nonlinear Equations

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} 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}. 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 points lie in a same cell and are therefore close to each other. 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