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.

Banach spaces and Ramsey Theory: some open problems

We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic flavor. The problems are exposed together with well known results related to them.Comment: 17 pages, no figures; RACSAM, to appea

Survey on the Tukey theory of ultrafilters

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page