Template-Type: ReDIF-Paper 1.0 Series: Tinbergen Institute Discussion Papers Creation-Date: 2007-10-31 Number: 07-084/1 Author-Name: Gerard van der Laan Author-Email: glaan@feweb.vu.nl Author-Workplace-Name: VU University Amsterdam Author-Name: Dolf Talman Author-Email: talman@uvt.nl Author-Workplace-Name: Tilburg University Author-Name: Zaifu Yang Author-Email: yang@ynu.ac.jp Author-Workplace-Name: Yokohama National University Title: Combinatorial Integer Labeling Theorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations Abstract: This discussion paper resulted in a publication in the 'Journal of Optimization Theory and Applications', 2010, 144, 391-407.

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. Classification-JEL: C61; C62; C68; C72; C58 Keywords: Sperner lemma; Tucker lemma; integer labeling; simplicial algorithm; discrete nonlinear equations File-Url: https://papers.tinbergen.nl/07084.pdf File-Format: application/pdf File-Size: 251686 bytes Handle: RePEc:tin:wpaper:20070084