• Corpus ID: 17922568

ON TOPOLOGICAL AND GEOMETRIC (194) CONFIGURATIONS JÜRGEN BOKOWSKI AND VINCENT PILAUD‡

@inproceedings{BokowskiONTA,
  title={ON TOPOLOGICAL AND GEOMETRIC (194) CONFIGURATIONS J{\"U}RGEN BOKOWSKI AND VINCENT PILAUD‡},
  author={J{\"u}rgen Bokowski and Vincent Pilaud}
}
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 on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of (n k) configurations for a given k has been subject to active research. A current front of research concerns geometric (n 4) configurations: it is now known that… 

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$(22_4)$ and $(26_4)$ configurations of lines
We present a technique to produce arrangements of lines with nice properties. As an application, we construct $(22_4)$ and $(26_4)$ configurations of lines. Thus concerning the existence of geometric
(224) and (264) Configurations of Lines
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  • Mathematics
    Ars Math. Contemp.
  • 2018
TLDR
A technique to produce arrangements of lines with nice properties and concerning the existence of geometric $(n_4)$ configurations, only the case $n=23$ remains open.

References

SHOWING 1-10 OF 10 REFERENCES
Connected ( n k ) configurations exist for almost all n
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
Enumerating topological $(n_k)$-configurations
TLDR
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.
) Configurations Exist for Almost All N –– an Update
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
On the finite set of missing geometric configurations (n4)
The combinatorial (19_4) configurations
TLDR
It is proved that two of the combinatorial (19 4 ) configurations are not geometrically realizable over any field, and the computation of the 971 171 combinatorsial (18 4) configurations which lacked an independent verification is confirmed.
There are no realizable 15_4- and 16_4-configurations.
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
Configurations of points and lines, volume 103 of Graduate Studies in Mathematics
  • Configurations of points and lines, volume 103 of Graduate Studies in Mathematics
  • 2009
Connected (n 4 ) configurations exist for almost all n—second update
  • Geombinatorics
  • 2006
Technische Universität Darmstadt E-mail address: juergen.bokowski@gmail.com (V. Pilaud) CNRS & LIX, ´ Ecole Polytechnique, Palaiseau E-mail address
  • Technische Universität Darmstadt E-mail address: juergen.bokowski@gmail.com (V. Pilaud) CNRS & LIX, ´ Ecole Polytechnique, Palaiseau E-mail address