The Borsuk-Ulam Theorem and Bisection of Necklaces
نویسندگان
چکیده
The Borsuk-Ulam theorem of topology is applied to a problem in discrete mathematics. A bisection of a necklace with k colors of beads is a collection of intervals whose union captures half the beads of each color. Every necklace with fc colors has a bisection formed by at most k cuts. Higherdimensional generalizations are considered.
منابع مشابه
Splitting Necklaces and a Generalization of the Borsuk-ulam Antipodal Theorem
We prove a very natural generalization of the Borsuk-Ulam antipodal Theorem and deduce from it, in a very straightforward way, the celebrated result of Alon [1] on splitting necklaces. Alon’s result says that t(k− 1) is an upper bound on the number of cutpoints of an opened t-coloured necklace so that the segments we get can be used to partition the set of vertices of the necklace into k subset...
متن کاملDigital Borsuk-Ulam theorem
The aim of this paper is to compute a simplicial cohomology group of some specific digital images. Then we define ringand algebra structures of a digital cohomology with the cup product. Finally, we prove a special case of the Borsuk-Ulam theorem fordigital images.
متن کاملA Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma
We show that Fan’s 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk–Ulam equivalents, we show that this lemma directly implies Sperner’s Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer fixed-point theorem, or that the Lustern...
متن کاملA definable Borsuk-Ulam type theorem
Let N = (R,+, ·, <, . . . ) be an o-minimal expansion of the standard structure of a real closed field R. A definably compact definable group G is a definable Borsuk-Ulam group if there exists an isovariant definable map f : V → W between representations of G, then dimV − dimV G ≤ dimW − dimW. We prove that if a finite group G satisfies the prime condition, then G is a definable Borsuk-Ulam gro...
متن کاملExtensions of the Borsuk-Ulam Theorem
Extensions of the Borsuk-Ulam Theorem by Timothy Prescott
متن کامل