Induced Ramsey-type results and binary predicates for point sets

Let A and B be two finite sets of points in the plane in general position (neither of these sets contains three collinear points). We say that A lies deep below B if every point from A lies below every line determined by two points from B and every point from B lies above every line determined by two points from A. A point set P is decomposable if either |P|=1 or there is a partition P₁∪P₂ of P into nonempty and decomposable sets such that P₁ is to the left of P₂ and P₁ is deep below P₂. Extending a result of Nešetřil and Valtr, we show that for every decomposable point set Q and a positive integer k there is a finite set P of points in the plane in general position that satisfies the following Ramsey-type statement. For any partition C₁∪⋯∪C_k of the pairs of points from P (that is, of the edges of the complete graph on P), there is a subset Q^′ of P with the same triple-orientations as Q such that all pairs of points from Q^′ are in the same part C_i. We then use this 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 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.