Consensus-halving via Theorems of Borsuk-ulam and Tucker Forest W. Simmons and Francis
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Consensus-halving via theorems of Borsuk-Ulam and Tucker
In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believes the portions are equal. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications o...
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This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tucker-property of a finite group G is introduced and its relation to the topological Borsuk-Ulam-property is discussed. Applications of the Tucker-property in combinatorics are demonstrated.
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The Borsuk-Ulam theorem and the Brouwer fixed point theorem are well-known theorems of topology with a very similar flavor. Both are non-constructive existence results with somewhat surprising conclusions. Most topology textbooks that cover these theorems (e.g., [4], [5], [6]) do not mention the two are related—although, in fact, the Borsuk-Ulam theorem implies the Brouwer Fixed Point Theorem. ...
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The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that discs and spheres can be substituted by large classes of manifolds with or without boundary.
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تاریخ انتشار 1999