On topological and geometric (194)configurations

Jürgen Bokowski; Vincent Pilaud

doi:10.1016/j.ejc.2015.03.008

On topological and geometric (19₄)configurations

Jürgen Bokowski
, Vincent Pilaud

Technische Universität Darmstadt

Research output: Contribution to journal › Article › peer-review

Abstract

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 (n₄)configurations: it is now known that geometric (n₄)configurations exist for all n≥18, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of (19₄)configurations: we obtain all topological (19₄)configurations among which none are geometrically realizable.

Original language	English
Pages (from-to)	4-17
Number of pages	14
Journal	European Journal of Combinatorics
Volume	50
DOIs	https://doi.org/10.1016/j.ejc.2015.03.008
Publication status	Published - 1 Nov 2015

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10.1016/j.ejc.2015.03.008

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title = "On topological and geometric (194)configurations",

abstract = "An (nk)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 (nk)configurations for a given k has been subject to active research. A current front of research concerns geometric (n4)configurations: it is now known that geometric (n4)configurations exist for all n≥18, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of (194)configurations: we obtain all topological (194)configurations among which none are geometrically realizable.",

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AB - An (nk)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 (nk)configurations for a given k has been subject to active research. A current front of research concerns geometric (n4)configurations: it is now known that geometric (n4)configurations exist for all n≥18, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of (194)configurations: we obtain all topological (194)configurations among which none are geometrically realizable.

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On topological and geometric (194)configurations

Abstract

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On topological and geometric (19₄)configurations