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Let $k$ and $p$ be positive integers and let $Q$ be a finite point set ingeneral position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is afinite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order typeand $\binom{Q'}{p}$ is monochromatic in $c$. Ne\v{s}et\v{r}il and Valtr provedthat for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They alsoproved that for every $k \ge 2$ and $p \ge 2$, there are point sets that arenot $(k,p)$-Ramsey.  As our main result, we introduce a new family of $(k,2)$-Ramsey point sets,extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new resultto show that for every $k$ there is a point set $P$ such that no function$\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size$k$ can satisfy the following "local consistency" property: if $\Gamma$ attainsthe same values on two ordered triples of points from $P$, then these tripleshave the same orientation. Intuitively, this implies that there cannot be sucha function that is defined locally and determines the orientation of pointtriples.

Induced Ramsey-type results and binary predicates for point sets