64 research outputs found
The Erd\H{o}s-Hajnal Conjecture for Paths and Antipaths
We prove that for every k, there exists such that every graph G on n
vertices not inducing a path and its complement contains a clique or a
stable set of size
A Note on "Regularity lemma for distal structures"
In a recent paper, Chernikov and Starchenko prove that graphs defined in
distal theories have strong regularity properties, generalizing previous
results about graphs defined by semi-algebraic relations. We give a shorter,
purely model-theoretic proof of this fact.Comment: 6 page
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
On the Yao-Yao partition theorem
The Yao-Yao partition theorem states that given a probability measure on an
affine space of dimension n having a density which is continuous and bounded
away from 0, it is possible to partition the space into 2^n regions of equal
measure in such a way that every affine hyperplane avoids at least one of the
regions. We give a constructive proof of this result and extend it to slightly
more general measures.Comment: 10 pages, file might be slightly different from the published versio
Density theorems for intersection graphs of t-monotone curves
A curve \gamma in the plane is t-monotone if its interior has at most t-1
vertical tangent points. A family of t-monotone curves F is \emph{simple} if
any two members intersect at most once. It is shown that if F is a simple
family of n t-monotone curves with at least \epsilon n^2 intersecting pairs
(disjoint pairs), then there exists two subfamilies F_1,F_2 \subset F of size
\delta n each, such that every curve in F_1 intersects (is disjoint to) every
curve in F_2, where \delta depends only on \epsilon. We apply these results to
find pairwise disjoint edges in simple topological graphs
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
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
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