Note on a Ramsey-type problem in geometry
type: Item
instance of: scholarly article
title: Note on a Ramsey-type problem in geometry
zbMATH Open document ID: 0801.52010
DOI: 10.1016/0097-3165(94)90025-6
author: György Csizmadia
author: Géza Tóth
published in: Journal of Combinatorial Theory. Series A
publication date: 1994-12-01T00:00:00Z
review text: Solving a problem of Erdős, \textit{R. Juhász} [J. Comb. Theory, Ser. A 27, 152-160 (1979; Zbl 0431.05001)] proved that given any colouring of the plane by two colours (red and blue), and a 4-point configuration \(K\) one can find either two red points at distance 1 from each other or a congruent copy of \(K\) all of whose points are blue. Juhász also proved that this theorem does not remain true for all configurations \(K\) with at least 12 points. Here this number 12 is reduced to 8 by considering the standard 2-colouring of the plane and taking for \(K\) the vertices of a suitable regular heptagon together with its center. (The standard 2- colouring is defined using the regular triangular lattice, where the minimum distance between two lattice points is 2. A point \(P\) is coloured red if and only if there is a lattice point whose distance from \(P\) is smaller than 1/2.) On the other hand, the authors show (by a density argument) that given any five-point configuration \(K\) in the plane, one can find even a translate of \(K\) all of whose vertices are blue in the standard 2-colouring. This supports the conjecture that the number 4 in Juhász' theorem may be replaced by 5.
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 52C10
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 51A20
zbMATH DE Number: 554402
zbMATH Keywords: Ramsey-type problems
zbMATH Keywords: point configurations
zbMATH Keywords: colouring of the plane
MaRDI profile type: Publication
full work available at URL: 0097-3165(94)90025-6
OpenAlex ID: W2021951258
cites work: Q3903002
cites work: Euclidean Ramsey theorems. I
cites work: Q4071751
cites work: Ramsey type theorems in the plane
cites work: Q5344631
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description: scientific article; zbMATH DE number 554402