Abstract
Let k and p be positive integers and let Q be a finite point set in general position in the plane. We say that Q is (k,p)-Ramsey if there is a finite point set P such that for every k-coloring c of (P choose p) there is a subset Q' of P such that Q' and Q have the same order type and (Q' choose p) is monochromatic in c.
Nešetřil and Valtr proved that for every k ELEMENT OF N, all point sets are (k,1)-Ramsey. They also proved that for every k GREATER-THAN OR EQUAL TO 2 and p GREATER-THAN OR EQUAL TO 2, there are point sets that are not (k,p)-Ramsey.
As our main result, we introduce a new family of (k,2)-Ramsey point sets, extending a result of Nešetřil and Valtr. We then use this new result to show that for every k there is a point set P such that no function Γ that maps ordered pairs of distinct points from P to a set of size k can satisfy the following "local consistency" property: if Γ attains the same values on two ordered triples of points from P, then these triples have the same orientation.
Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.