Consensus-halving via theorems of Borsuk-Ulam and Tucker

Consensus-halving via Theorems of Borsuk-ulam and Tucker Forest W. Simmons and Francis

The Borsuk-Ulam-property, Tucker-property and constructive proofs in combinatorics

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.

Extensions of Sperner and Tucker's lemma for manifolds

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.

Borsuk-Ulam Implies Brouwer: A Direct Construction

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. ...

Theorems of Borsuk-ulam Type for Flats and Common Transversals

In this paper some results on the topology of the space of k-flats in Rn are proved, similar to the Borsuk-Ulam theorem on coverings of sphere. Some corollaries on common transversals for families of compact sets in Rn, and on measure partitions by hyperplanes, are deduced.