673 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
Lower bounds on geometric Ramsey functions
We continue a sequence of recent works studying Ramsey functions for
semialgebraic predicates in . A -ary semialgebraic predicate
on is a Boolean combination of polynomial
equations and inequalities in the coordinates of points
. A sequence of points in
is called -homogeneous if either holds for all choices , or it
holds for no such choice. The Ramsey function is the smallest
such that every point sequence of length contains a -homogeneous
subsequence of length .
Conlon, Fox, Pach, Sudakov, and Suk constructed the first examples of
semialgebraic predicates with the Ramsey function bounded from below by a tower
function of arbitrary height: for every , they exhibit a -ary
in dimension with bounded below by a tower of height .
We reduce the dimension in their construction, obtaining a -ary
semialgebraic predicate on with bounded
below by a tower of height .
We also provide a natural geometric Ramsey-type theorem with a large Ramsey
function. We call a point sequence in order-type homogeneous
if all -tuples in have the same orientation. Every sufficiently long
point sequence in general position in contains an order-type
homogeneous subsequence of length , and the corresponding Ramsey function
has recently been studied in several papers. Together with a recent work of
B\'ar\'any, Matou\v{s}ek, and P\'or, our results imply a tower function of
of height as a lower bound, matching an upper bound by Suk up
to the constant in front of .Comment: 12 page
-permutability and linear Datalog implies symmetric Datalog
We show that if is a core relational structure such that
CSP() can be solved by a linear Datalog program, and is
-permutable for some , then CSP() can be solved by a symmetric
Datalog program (and thus CSP() lies in deterministic logspace). At
the moment, it is not known for which structures will CSP() be solvable by a linear Datalog program. However, once somebody obtains a
characterization of linear Datalog, our result immediately gives a
characterization of symmetric Datalog
Finite-Degree Predicates and Two-Variable First-Order Logic
We consider two-variable first-order logic on finite words with a fixed
number of quantifier alternations. We show that all languages with a neutral
letter definable using the order and finite-degree predicates are also
definable with the order predicate only. From this result we derive the
separation of the alternation hierarchy of two-variable logic on this
signature
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