• Corpus ID: 14220441

# 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…
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## References

SHOWING 1-2 OF 2 REFERENCES
Which (n 4 ) Configurations Exist ?
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.
Astral (n 4 ) Configurations
A family of n points and n (straight) lines in the Euclidean plane is said to be an (n 4) configuration provided each point is on four of the lines and each line contains four of the points. A