Note on a Ramsey-type problem in geometry (Q1320393)

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.