Boris Bukh, Jiří Matoušek

Duke Math. J. 163 (12), 2243-2270, (15 September 2014) DOI: 10.1215/00127094-2785915
KEYWORDS: 05D10, 52C45

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 ${\mathbb{R}}^{d}$.