Connected ( n k ) configurations exist for almost all n
@inproceedings{Grnbaum2021ConnectedN, title={Connected ( n k ) configurations exist for almost all n}, author={Branko Gr{\"u}nbaum}, year={2021} }
A geometric (nk) configuration is a collection of points and straight lines, typically in the Euclidean plane, so that each line passes through k of the points and each of the points lies on k of the lines. In a series of papers, Branko Grünbaum showed that geometric (n4) configurations exist for all n ≥ 24, using a series of geometric constructions later called the “Grünbaum Calculus”. In this talk, we will show that for each k > 4, there exists an integer Nk so that for all n ≥ Nk, there…
12 Citations
Connected geometric (n_k) configurations exist for almost all n
- MathematicsThe Art of Discrete and Applied Mathematics
- 2021
In a series of papers and in his 2009 book on configurations Branko Grunbaum described a sequence of operations to produce new (n4) configurations from various input configurations. These operations…
ON TOPOLOGICAL AND GEOMETRIC (194) CONFIGURATIONS JÜRGEN BOKOWSKI AND VINCENT PILAUD‡
- Mathematics
An (n k) configuration is a set of n points and n lines such that each point lies on k lines while each line contains k points. The configuration is geometric, topological, or combinatorial depending…
) Configurations Exist for Almost All N –– an Update
- Mathematics
An (n k) configuration is a family of n points and n (straight) lines in the Euclidean plane such that each point is on precisely k of the lines, and each line contains precisely k of the points. A…
Eventually, geometric $(n_{k})$ configurations exist for all $n$
- Mathematics
- 2021
In a series of papers and in his 2009 book on configurations Branko Grünbaum described a sequence of operations to produce new (n4) configurations from various input configurations. These operations…
On the generation of topological (nk)-configurations
- MathematicsCCCG
- 2011
This work provides an algorithm for generating all combinatorial (nk)-congurations that admit a topological realization, for given n and k, and enables us to conrm, in just one hour with a Java code of the second author, a satisability result of Lars Schewe in [11], obtained after several months of CPU-time.
Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces
- MathematicsArs Math. Contemp.
- 2018
It is shown that every incidence structure (and therefore also every combinatorial configuration) can be realized as a monotone quasiline arrangement in the real projective plane.
Enumerating topological $(n_k)$-configurations
- MathematicsCCCG 2011
- 2011
This work provides an algorithm for generating, for given $n and $k$, all topological $(n_k)$-configurations up to combinatorial isomorphism, without enumerating first all combinatorsial $(n-k) $- configurations.
There are no realizable 15_4- and 16_4-configurations.
- Mathematics
- 2005
There exist a finite number of natural numbers n for which we do not know whether a realizable n4-configuration does exist. We settle the two smallest unknown cases n = 15 and n = 16. In these cases…
Quasi-configurations: building blocks for point-line configurations
- MathematicsArs Math. Contemp.
- 2016
The motivation is the problem of the existence of $(n_4)$ configurations, still open for few remaining values of $n, which is based on quasi-configurations: point-line incidence structures where each point is incident to at least $3$ lines and each line is incidentto at least$3$ points.
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An (n k) configuration is a family of n points and n (straight) lines in the Euclidean plane such that each point is on precisely k of the lines, and each line contains precisely k of the points.…
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