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Borsuk and Ramsey type questions in Euclidean space
- Source :
- Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham, Cambridge University Press, 2018, 259--277
- Publication Year :
- 2017
-
Abstract
- We give a short survey of problems and results on (1) diameter graphs and hypergraphs, and (2) geometric Ramsey theory. We also make some modest contributions to both areas. Extending a well known theorem of Kahn and Kalai which disproved Borsuk's conjecture, we show that for any integer $r\ge 2$, there exist $\varepsilon=\varepsilon(r)>0$ and $d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a finite point set $P\subset\mathbb{R}^d$ of diameter $1$ such that no matter how we color the elements of $P$ with fewer than $(1+\varepsilon)^{\sqrt{d}}$ colors, we can always find $r$ points of the same color, any two of which are at distance $1$.<br />Comment: second version addresses changes arising from the referee reports
- Subjects :
- Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Journal :
- Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham, Cambridge University Press, 2018, 259--277
- Publication Type :
- Report
- Accession number :
- edsarx.1702.03707
- Document Type :
- Working Paper