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

Gerard van der Laan, Dolf Talman () and Z.F. Yang
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Z.F. Yang: Tilburg University, Center for Economic Research

No 2007-88, Discussion Paper from Tilburg University, Center for Economic Research

Abstract: 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 f?1;?2; ? ? ? ;?ng 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 f?1;?2; ? ? ? ;?ng. 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 set f0; 1gn+q for some integral vector q. 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.

JEL-codes: C61 C62 C68 C72 (search for similar items in EconPapers)
Date: 2007
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Working Paper: Combinatorial Integer Labeling Theorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations (2007)
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