K. Borsuk, Drei Satze  uber die n-dimensionale euklidische Sphare, Fund. Math., 20 (1933), 177-190.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....cover in R 3 . Moreover U 3 is the intersection of the 6 strips corresponding to the 6 edges of a regular tetrahedron of edge length 1, which is a rhombic dodecahedron with distance of opposite faces equal to 1. A frequent application of universal covers is in the so called Borsuk problem [Bor]: if X R n has diameter 1, can it be decomposed into n 1 sets X 1 ; X n 1 of smaller diameters We note that for all suciently large n Borsuk s problem has a negative solution [KhKa] even for nite sets X , but the smallest n, for which a counterexample is known, is n = 561, cf. ....

K. Borsuk, Drei Satze  uber die n-dimensionale euklidische Sphare, Fund. Math. 20 (1933), 177-190, Zbl. 6,424,03

..... In particular, much attention have been paid to the sphere S n 1 of radius 1 in IR n . Consider H = 2] for 0 2 and the distance graph that H de nes on S n 1 . This graph, the Borsuk graph, was introduced and studied by Erd os and A. Hajnal [25] It follows from Borsuk s Theorem [6] (and in fact, is equivalent to 9 it) that H (S n 1 ) n 1. Combining this with the Erd os de Bruijn Theorem [9] on the chromatic number of in nite graphs, one obtains one of the earliest constructions for nite graphs with high chromatic number and high odd girth. Erd os and R. L. Graham ....

K. Borsuk, Drei Satze  uber die n-dimensionale euklidische Sphare, Fund. Math. 20 (1933), 177-190.

....relates the 0=1 case of Borsuk s problem to the coloring problem for the Hamming graphs, to the geometry of a Hamming code, as well as to some upper bounds for the sizes of binary codes. 1 Introduction The Borsuk conjecture is a puzzling problem: posed in 1933, in the famous paper by K. Borsuk [6] that contained the Borsuk Ulam theorem, it has resisted all attempts of proof until in 1992 Kahn and Kalai [11] announced that the conjecture is false, due to counterexamples in dimensions 1325 and higher. After much subsequent work, we now know that the Borsuk conjecture is false in all ....

K. Borsuk: Drei Satze  uber die n-dimensionale euklidische Sphare, Fundamenta Math. 20 (1933), 177-190.

....lie opposite each other on the sphere i.e. fx; xg for some x. The Borsuk Ulam Theorem. Let f : S n R n be a continuous map. There exists a pair of antipodal points on S n that are mapped by f to the same point in R n . This theorem was conjectured by S. Ulam and proved by K. Borsuk [1] in 1933. In particular, it says that if f = f 1 ; f 2 ; fn ) is a set of n continuous real valued functions on the sphere, then there must be antipodal points on which all the functions agree. For instance, one interpretation for the case n = 2 is that there is always a pair of antipodal ....

K. Borsuk, Drei Satze  uber die n-dimensionale euklidische Sphare, Fund. Math., 20 (1933), 177-190.

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