Enumerating Motzkin–Rabin geometries

Wamelen, Paul van

doi:10.1002/jcd.20119

Cited by 1 publication

(4 citation statements)

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“…It is known that a non-collinear SGC in F 2 (x) 2 has at least 15 points (by the results in [29]), and a non-collinear MRC in P 2 (F 2 (x)) needs at least 19 points [45]. By Lemma 13 a non-collinear 4-SGC over F 2 (x) needs at least 22 points.…”

Section: Proposition 17mentioning

confidence: 92%

“…By the results of Kelly and Nwankpa [29] it can be seen that if D is rootfree and of characteristic 0, then a 2-dimensional SGC in P n (D) must have at least 15 points. There exist root-free fields of characteristic 0 admitting non-collinear SGCs, e.g., Q( √ −7) [45]. This is the only known example of a non-collinear SGC over C that is not obtained from a root of unity via Proposition 12.…”

Section: Proposition 17mentioning

confidence: 99%

“…Kelly and Nwankpa [29] also determined which of the SG geometries on at most 14 points are embeddable in Desarguesian projective planes or spaces. The MR geometries on up to 15 points have been enumerated in [37], and up to 18 points in [45] (Table 1).…”

Section: Small Chromatic Geometriesmentioning

confidence: 99%

“…The field in the following problem is also root-free. It is known that if there exists a two-dimensional MRC in P n (C) then it must have at least 19 points [37,45] By Theorem C, a configuration giving a counterexample to the above question would have at least 51 points. It would also be interesting to find any non-commutative division ring of characteristic 0 with a non-coplanar SGC.…”

Section: Proposition 17mentioning

confidence: 99%

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The Sylvester–Gallai theorem, colourings and algebra

Pretorius

Swanepoel

2009

Discrete Mathematics

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Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem:Let S be a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct points A, B ∈ S sharing a colour there is a third point C ∈ S, of the other colour, collinear with A and B. Then all the points in S are collinear.We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A, B ∈ S sharing a colour there is a third point C ∈ S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly two-coloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly's complex Sylvester-Gallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a three-dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex Sylvester-Gallai theorem.

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Section: Proposition 17mentioning

confidence: 92%

Section: Proposition 17mentioning

confidence: 99%