5 research outputs found
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
Curves in R^d intersecting every hyperplane at most d+1 times
By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a
closed interval. We call a curve gamma in R^d at most k crossing if it
intersects every hyperplane at most k times (counted with multiplicity). The at
most d crossing curves in R^d are often called convex curves and they form an
important class; a primary example is the moment curve
{(t,t^2,...,t^d):t\in[0,1]}. They are also closely related to Chebyshev
systems, which is a notion of considerable importance, e.g., in approximation
theory. We prove that for every d there is M=M(d) such that every at most d+1
crossing curve in R^d can be subdivided into at most M convex curves. As a
consequence, based on the work of Elias, Roldan, Safernova, and the second
author, we obtain an essentially tight lower bound for a geometric Ramsey-type
problem in R^d concerning order-type homogeneous sequences of points,
investigated in several previous papers.Comment: Corrected proof of Lemma 3.