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Tag Archives: Borsuk’s conjecture
Around Borsuk’s Conjecture 3: How to Save Borsuk’s conjecture
Borsuk asked in 1933 if every bounded set K of diameter 1 in can be covered by d+1 sets of smaller diameter. A positive answer was referred to as the “Borsuk Conjecture,” and it was disproved by Jeff Kahn and me in 1993. … Continue reading
Andriy Bondarenko Showed that Borsuk’s Conjecture is False for Dimensions Greater Than 65!
The news in brief Andriy V. Bondarenko proved in his remarkable paper The Borsuk Conjecture for two-distance sets that the Borsuk’s conjecture is false for all dimensions greater than 65. This is a substantial improvement of the earlier record (all dimensions … Continue reading
A Weak Form of Borsuk Conjecture
Problem: Let P be a polytope in with n facets. Is it always true that P can be covered by n sets of smaller diameter? I also asked this question over mathoverflow, with some background and motivation.
Around Borsuk’s Conjecture 1: Some Problems
Greetings to all! Karol Borsuk conjectured in 1933 that every bounded set in can be covered by sets of smaller diameter. In a previous post I described the counterexample found by Jeff Kahn and me. I will devote a few posts … Continue reading
The Combinatorics of Cocycles and Borsuk’s Problem.
Cocycles Definition: A -cocycle is a collection of -subsets such that every -set contains an even number of sets in the collection. Alternative definition: Start with a collection of -sets and consider all -sets that contain an odd number of members … Continue reading
Raigorodskii’s Theorem: Follow Up on Subsets of the Sphere without a Pair of Orthogonal Vectors
Andrei Raigorodskii (This post follows an email by Aicke Hinrichs.) In a previous post we discussed the following problem: Problem: Let be a measurable subset of the -dimensional sphere . Suppose that does not contain two orthogonal vectors. How large … Continue reading
A Little Story Regarding Borsuk’s Conjecture
Jeff Kahn Jeff and I worked on the problem for several years. Once he visited me with his family for two weeks. Before the visit I emailed him and asked: What should we work on in your visit? Jeff asnwered: … Continue reading
Borsuk’s Conjecture
Karol Borsuk conjectured in 1933 that every bounded set in can be covered by sets of smaller diameter. Jeff Kahn and I found a counterexample in 1993. It is based on the Frankl-Wilson theorem. Let be the set of vectors of length . … Continue reading
