2,193 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
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
A note on order-type homogeneous point sets
Let OT_d(n) be the smallest integer N such that every N-element point
sequence in R^d in general position contains an order-type homogeneous subset
of size n, where a set is order-type homogeneous if all (d+1)-tuples from this
set have the same orientation. It is known that a point sequence in R^d that is
order-type homogeneous forms the vertex set of a convex polytope that is
combinatorially equivalent to a cyclic polytope in R^d. Two famous theorems of
Erdos and Szekeres from 1935 imply that OT_1(n) = Theta(n^2) and OT_2(n) =
2^(Theta(n)). For d \geq 3, we give new bounds for OT_d(n). In particular:
1. We show that OT_3(n) = 2^(2^(Theta(n))), answering a question of
Eli\'a\v{s} and Matou\v{s}ek.
2. For d \geq 4, we show that OT_d(n) is bounded above by an exponential
tower of height d with O(n) in the topmost exponent
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