## Duke Mathematical Journal

### 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 $\boldsymbol {\Phi }=\{x_{1}\lt x_{2},x_{1}\ge x_{2}\}$ is Erdős–Szekeres with Ramsey function $\operatorname {ES}_{\boldsymbol {\Phi }}(n)=(n-1)^{2}+1$.

In general, we consider an arbitrary finite set $\boldsymbol {\Phi }=\{\Phi_{1},\ldots,\Phi_{m}\}$ of semialgebraic predicates, meaning that each $\Phi_{j}=\Phi_{j}(x_{1},\ldots,x_{k})$ is a Boolean combination of polynomial equations and inequalities in some number $k$ of real variables. We define $\boldsymbol {\Phi }$ 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 $\Phi_{j}(b_{i_{1}},\ldots,b_{i_{k}})$ holds for every choice of indices $i_{1},i_{2},\ldots,i_{k}$, $1\le i_{1}\lt i_{2}\lt \cdots\lt i_{k}\le n$. We write $\operatorname {ES}_{\boldsymbol {\Phi }}(n)$ 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 $\boldsymbol {\Phi }$ that is Erdős–Szekeres, there is a constant $C$ such that $\operatorname {ES}_{\boldsymbol {\Phi }}(n)\le2^{2^{Cn}}$. Second, there is an algorithm that, given $\boldsymbol {\Phi }$, 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 $\mathbb {R}^{d}$.

#### Article information

Source
Duke Math. J., Volume 163, Number 12 (2014), 2243-2270.

Dates
First available in Project Euclid: 15 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1410789515

Digital Object Identifier
doi:10.1215/00127094-2785915

Mathematical Reviews number (MathSciNet)
MR3263034

Zentralblatt MATH identifier
1301.05351

#### Citation

Bukh, Boris; Matoušek, Jiří. Erdős–Szekeres-type statements: Ramsey function and decidability in dimension $1$. Duke Math. J. 163 (2014), no. 12, 2243--2270. doi:10.1215/00127094-2785915. https://projecteuclid.org/euclid.dmj/1410789515