Induced Ramsey-Type Results and Binary Predicates for Point Sets

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 p ) there is a subset Q′ of P such that Q′ and Q have the same order type and ( Q′ p ) is monochromatic in c. Nešetřil and Valtr proved that for every k ∈ 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.