  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 $\binom{P}{p}$
there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type
and $\binom{Q'}{p}$ is monochromatic in $c$. Ne\v{s}et\v{r}il and Valtr proved
that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also
proved that for every $k \ge 2$ and $p \ge 2$, 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 Ne\v{s}et\v{r}il and Valtr. We then use this new result
to 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 $\Gamma$ 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.
