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.
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Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number
2007-88.
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