4 research outputs found
Induced Ramsey-type results and binary predicates for point sets
Let and be positive integers and let be a finite point set in
general position in the plane. We say that is -Ramsey if there is a
finite point set such that for every -coloring of
there is a subset of such that and have the same order type
and is monochromatic in . Ne\v{s}et\v{r}il and Valtr proved
that for every , all point sets are -Ramsey. They also
proved that for every and , there are point sets that are
not -Ramsey.
As our main result, we introduce a new family of -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 there is a point set such that no function
that maps ordered pairs of distinct points from to a set of size
can satisfy the following "local consistency" property: if attains
the same values on two ordered triples of points from , 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.Comment: 22 pages, 3 figures, final version, minor correction