Induced Ramsey-type results and binary predicates for point sets

Let k and p be positive integers and let Q be a nite point set in general position in the plane. We say that Q is (k; p)-Ramsey if there is a nite point set P such that for every k-coloring c of (Formula presented) there is a subset Q′ of P such that Q′ and Q have the same order type and (Formula presented) is monochromatic in c. Nešetřil and Valtr proved that for every k ࢠ N, all point sets are (k, 1)-Ramsey. They also proved that for every k ≥ 2 and p ≥ 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.