109 research outputs found
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.
Topological lower bounds for the chromatic number: A hierarchy
This paper is a study of ``topological'' lower bounds for the chromatic
number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978,
in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology.
This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with
all -element subsets of as vertices and all pairs of
disjoint sets as edges, has chromatic number . Several other proofs
have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz,
Greene, and others), all of them based on some version of the Borsuk--Ulam
theorem, but otherwise quite different. Each can be extended to yield some
lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe
that \emph{every} finite graph may be represented as a generalized Kneser
graph, to which the above bounds apply.)
We show that these bounds are almost linearly ordered by strength, the
strongest one being essentially Lov\'asz' original bound in terms of a
neighborhood complex. We also present and compare various definitions of a
\emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz
and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but
the construction is simpler and functorial, mapping graphs with homomorphisms
to -spaces with -maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea
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.
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