Erdős–Szekeres-type statements: Ramsey function and decidability in dimension 1

Abstract

A classical and widely used lemma of Erdős and Szekeres asserts that for every $n$ there exists $N$ such that every $N$-term sequence $\underline{a}$ of real numbers contains an $n$-term increasing subsequence or an $n$-term nonincreasing 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 $\mathit{\Phi }=\{x_1<x_2,x_1\ge x_2\}$ is Erdős–Szekeres with Ramsey function ${ES}_{\mathit{\Phi }}\left(n\right)=(n-1{)}^2+1$.

In general, we consider an arbitrary finite set $\mathit{\Phi }=\{{\Phi }_1,\dots ,{\Phi }_m\}$ of semialgebraic predicates, meaning that each ${\Phi }_j={\Phi }_j(x_1,\dots ,x_k)$ is a Boolean combination of polynomial equations and inequalities in some number $k$ of real variables. We define $\mathit{\Phi }$ to be Erdős–Szekeres if for every $n$ there exists $N$ such that each $N$-term sequence $\underline{a}$ of real numbers has an $n$-term subsequence $\underline{b}$ such that at least one of the ${\Phi }_j$ holds everywhere on $\underline{b}$, which means that ${\Phi }_j(b_{i_1},\dots ,b_{i_k})$ holds for every choice of indices $i_1,i_2,\dots ,i_k$, $1\le i_1<i_2<\cdots <i_k\le n$. We write ${ES}_{\mathit{\Phi }}\left(n\right)$ 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 $\mathit{\Phi }$ that is Erdős–Szekeres, there is a constant $C$ such that ${ES}_{\mathit{\Phi }}\left(n\right)\le 2^{2^{Cn}}$. Second, there is an algorithm that, given $\mathit{\Phi }$, decides whether it is Erdős–Szekeres; thus, $1$-dimensional Erdős–Szekeres-style theorems can in principle be proved automatically.

We regard these results as a starting point in investigating analogous questions for $d$-dimensional predicates, where instead of sequences of real numbers, we consider sequences of points in ${\mathbb{R}}^d$ (and semialgebraic predicates in their coordinates). This setting includes many results and problems in geometric Ramsey theory, and it appears considerably more involved. Here we prove a decidability result for algebraic predicates in ${\mathbb{R}}^d$ (i.e., conjunctions of polynomial equations), as well as for a multipartite version of the problem with arbitrary semialgebraic predicates in ${\mathbb{R}}^d$.