Some old and new problems in combinatorial geometry. I: Around Borsuk’s problem. (English) Zbl 1361.51008
Czumaj, Artur (ed.) et al., Surveys in combinatorics 2015. Papers based on the 25th British combinatorial conference, University of Warwick, Coventry, UK, July 6–10, 2015. Cambridge: Cambridge University Press (ISBN 978-1-107-46250-2/pbk; 978-1-316-10685-3/ebook). London Mathematical Society Lecture Note Series 424, 147-174 (2015).
This is a very interesting article forming a broad and deep collection of open problems and conjectures related to the Borsuk conjecture from [K. Borsuk, Fundam. Math. 20, 177–190 (1933; Zbl 0006.42403)] and its later disproof by J. Kahn and the author [Bull. Am. Math. Soc., New Ser. 29, No.1, 60–62 (1993; Zbl 0786.52002)] using results by P. Frankl and R. M. Wilson [Combinatorica 1, 357–368 (1981; Zbl 0498.05048)].
The Borsuk conjecture stated that each set of diameter 1 in \(\mathbb{R}^d\) can be covered by \(d+1\) sets of smaller diameter. The questions posed in the present article address variations and weakenings of the conjecture, bounds on the number of coverings, extremal set and spherical set special cases of the conjecture, cases for which the conjecture does hold, related graph and complex questions, the conjecture posed for particular metrics, and questions relating to the counterexamples, in particular to cocycles and Turán numbers.
For the entire collection see [Zbl 1348.05003].
The Borsuk conjecture stated that each set of diameter 1 in \(\mathbb{R}^d\) can be covered by \(d+1\) sets of smaller diameter. The questions posed in the present article address variations and weakenings of the conjecture, bounds on the number of coverings, extremal set and spherical set special cases of the conjecture, cases for which the conjecture does hold, related graph and complex questions, the conjecture posed for particular metrics, and questions relating to the counterexamples, in particular to cocycles and Turán numbers.
For the entire collection see [Zbl 1348.05003].
Reviewer: Thomas Britz (Sydney)
MSC:
| 51D20 | Combinatorial geometries and geometric closure systems |
