A Ramsey property of order types
The main result of this paper is that for any finite set \(A\) of points in the plane, any fixed number \(k\) of colors, and any distance bound \(\varepsilon > 0\) there is a finite set \(B\) in the plane such that in any \(k\)-coloring of \(B\) there is a monochromatic subset \(C\) of \(B\) that is induced by its convex hull in \(B\) and that is an affine image of an \(\varepsilon\)-approximation of \(A\) in the Hausdorff sense. This implies a number of Ramsey-type theorems on the existence of subsets of a given order type, as well as on the distances forced by any realization of a given order type.
scientific article; zbMATH DE number 1123549