58 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
Colouring versus density in integers and Hales-Jewett cubes
We construct for every integer and every real a set of integers which, when coloured with
finitely many colours, contains a monochromatic -term arithmetic
progression, whilst every finite has a subset of
size that is free of arithmetic progressions of length .
This answers a question of Erd\H{o}s, Ne\v{s}et\v{r}il, and the second author.
Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett
version of this result.Comment: 5 figure
On globally sparse Ramsey graphs
We say that a graph has the Ramsey property w.r.t.\ some graph and
some integer , or is -Ramsey for short, if any -coloring
of the edges of contains a monochromatic copy of . R{\"o}dl and
Ruci{\'n}ski asked how globally sparse -Ramsey graphs can possibly
be, where the density of is measured by the subgraph with
the highest average degree. So far, this so-called Ramsey density is known only
for cliques and some trivial graphs . In this work we determine the Ramsey
density up to some small error terms for several cases when is a complete
bipartite graph, a cycle or a path, and colors are available
Edge-ordered Ramsey numbers
We introduce and study a variant of Ramsey numbers for edge-ordered graphs,
that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey
number of an edge-ordered graph
is the minimum positive integer such that there exists an edge-ordered
complete graph on vertices such that every 2-coloring of
the edges of contains a monochromatic copy of
as an edge-ordered subgraph of .
We prove that the edge-ordered Ramsey number
is finite for every edge-ordered graph and we obtain better
estimates for special classes of edge-ordered graphs. In particular, we prove
for every bipartite
edge-ordered graph on vertices. We also introduce a natural
class of edge-orderings, called lexicographic edge-orderings, for which we can
prove much better upper bounds on the corresponding edge-ordered Ramsey
numbers.Comment: Minor revision, 16 pages, 1 figure. An extended abstract of this
paper will appeared in the Eurocomb 2019 proceedings in Acta Mathematica
Universitatis Comenianae. The paper has been accepted to the European Journal
of Combinatoric
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