4,320 research outputs found
Semi-algebraic Ramsey numbers
Given a finite point set , a -ary semi-algebraic
relation on is the set of -tuples of points in , which is
determined by a finite number of polynomial equations and inequalities in
real variables. The description complexity of such a relation is at most if
the number of polynomials and their degrees are all bounded by . The Ramsey
number is the minimum such that any -element point set
in equipped with a -ary semi-algebraic relation , such
that has complexity at most , contains members such that every
-tuple induced by them is in , or members such that every -tuple
induced by them is not in .
We give a new upper bound for for and fixed.
In particular, we show that for fixed integers , establishing a subexponential upper bound on .
This improves the previous bound of due to Conlon, Fox, Pach,
Sudakov, and Suk, where is a very large constant depending on and
. As an application, we give new estimates for a recently studied
Ramsey-type problem on hyperplane arrangements in . We also study
multi-color Ramsey numbers for triangles in our semi-algebraic setting,
achieving some partial results
Semi-algebraic and semi-linear Ramsey numbers
An -uniform hypergraph is semi-algebraic of complexity
if the vertices of correspond to points in
, and the edges of are determined by the sign-pattern of
degree- polynomials. Semi-algebraic hypergraphs of bounded complexity
provide a general framework for studying geometrically defined hypergraphs.
The much-studied semi-algebraic Ramsey number
denotes the smallest such that every -uniform semi-algebraic hypergraph
of complexity on vertices contains either a clique of size
, or an independent set of size . Conlon, Fox, Pach, Sudakov, and Suk
proved that R_{r}^{\mathbf{t}}(n,n)<\mbox{tw}_{r-1}(n^{O(1)}), where
\mbox{tw}_{k}(x) is a tower of 2's of height with an on the top. This
bound is also the best possible if is sufficiently large with
respect to . They conjectured that in the asymmetric case, we have
for fixed . We refute this conjecture by
showing that for some
complexity .
In addition, motivated by results of Bukh-Matou\v{s}ek and
Basit-Chernikov-Starchenko-Tao-Tran, we study the complexity of the Ramsey
problem when the defining polynomials are linear, that is, when . In
particular, we prove that , while
from below, we establish .Comment: 23 pages, 1 figur
Ramsey-Turan numbers for semi-algebraic graphs
A semi-algebraic graph G = (V, E) is a graph where the vertices are points in R-d, and the edge set E is defined by a semi-algebraic relation of constant complexity on V. In this note, we establish the following Ramsey-Turan theorem: for every integer p >= 3, every K-p-free semi-algebraic graph on n vertices with independence number o(n) has at most 1/2(1 - 1/inverted right perpendicularp/2inverted left perpendicular - 1 + o(1)) n(2) edges. Here, the dependence on 1-1 the complexity of the semi-algebraic relation is hidden in the o(1) term. Moreover, we show that this bound is tight
Semi-algebraic colorings of complete graphs
We consider -colorings of the edges of a complete graph, where each color
class is defined semi-algebraically with bounded complexity. The case
was first studied by Alon et al., who applied this framework to obtain
surprisingly strong Ramsey-type results for intersection graphs of geometric
objects and for other graphs arising in computational geometry. Considering
larger values of is relevant, e.g., to problems concerning the number of
distinct distances determined by a point set.
For and , the classical Ramsey number is the
smallest positive integer such that any -coloring of the edges of ,
the complete graph on vertices, contains a monochromatic . It is a
longstanding open problem that goes back to Schur (1916) to decide whether
, for a fixed . We prove that this is true if each color
class is defined semi-algebraically with bounded complexity. The order of
magnitude of this bound is tight. Our proof is based on the Cutting Lemma of
Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for
multicolored semi-algebraic graphs, which is of independent interest. The same
technique is used to address the semi-algebraic variant of a more general
Ramsey-type problem of Erd\H{o}s and Shelah
Erdos-Szekeres-type statements: Ramsey function and decidability in dimension 1
A classical and widely used lemma of Erdos and Szekeres asserts that for
every n there exists N such that every N-term sequence a of real numbers
contains an n-term increasing subsequence or an n-term nondecreasing
subsequence; quantitatively, the smallest N with this property equals
(n-1)^2+1. In the setting of the present paper, we express this lemma by saying
that the set of predicates Phi={x_1<x_2,x_1\ge x_2}$ is Erdos-Szekeres with
Ramsey function ES_Phi(n)=(n-1)^2+1.
In general, we consider an arbitrary finite set Phi={Phi_1,...,Phi_m} of
semialgebraic predicates, meaning that each Phi_j=Phi_j(x_1,...,x_k) is a
Boolean combination of polynomial equations and inequalities in some number k
of real variables. We define Phi to be Erdos-Szekeres if for every n there
exists N such that each N-term sequence a of real numbers has an n-term
subsequence b such that at least one of the Phi_j holds everywhere on b, which
means that Phi_j(b_{i_1},...,b_{i_k}) holds for every choice of indices
i_1,i_2,...,i_k, 1<=i_1<i_2<... <i_k<= n. We write ES_Phi(n) for the smallest N
with the above property.
We prove two main results. First, the Ramsey functions in this setting are at
most doubly exponential (and sometimes they are indeed doubly exponential): for
every Phi that is Erd\H{o}s--Szekeres, there is a constant C such that
ES_Phi(n) < exp(exp(Cn)). Second, there is an algorithm that, given Phi,
decides whether it is Erdos-Szekeres; thus, one-dimensional
Erdos-Szekeres-style theorems can in principle be proved automatically.Comment: minor fixes of the previous version. to appear in Duke Math.
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
Ramsey-type theorems for lines in 3-space
We prove geometric Ramsey-type statements on collections of lines in 3-space.
These statements give guarantees on the size of a clique or an independent set
in (hyper)graphs induced by incidence relations between lines, points, and
reguli in 3-space. Among other things, we prove that: (1) The intersection
graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}).
(2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all
stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no
6-subset is stabbed by one line. (3) Every set of n lines in general position
in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a
subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus.
The proofs of these statements all follow from geometric incidence bounds --
such as the Guth-Katz bound on point-line incidences in R^3 -- combined with
Tur\'an-type results on independent sets in sparse graphs and hypergraphs.
Although similar Ramsey-type statements can be proved using existing generic
algebraic frameworks, the lower bounds we get are much larger than what can be
obtained with these methods. The proofs directly yield polynomial-time
algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi
Combinatorial complexity in o-minimal geometry
In this paper we prove tight bounds on the combinatorial and topological
complexity of sets defined in terms of definable sets belonging to some
fixed definable family of sets in an o-minimal structure. This generalizes the
combinatorial parts of similar bounds known in the case of semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the applicability of
results on combinatorial and topological complexity of arrangements studied in
discrete and computational geometry. As a sample application, we extend a
Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic
sets of fixed description complexity to this more general setting.Comment: 25 pages. Revised version. To appear in the Proc. London Math. So
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