1,680 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
On path-quasar Ramsey numbers
Let and be two given graphs. The Ramsey number is
the least integer such that for every graph on vertices, either
contains a or contains a . Parsons gave a recursive
formula to determine the values of , where is a path on
vertices and is a star on vertices. In this note, we first
give an explicit formula for the path-star Ramsey numbers. Secondly, we study
the Ramsey numbers , where is a linear forest on
vertices. We determine the exact values of for the cases
and , and for the case that has no odd component.
Moreover, we give a lower bound and an upper bound for the case and has at least one odd component.Comment: 7 page
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