A classical and widely used lemma of Erdős and Szekeres asserts that for every there exists such that every -term sequence of real numbers contains an -term increasing subsequence or an -term nonincreasing subsequence; quantitatively, the smallest with this property equals . In the setting of the present paper, we express this lemma by saying that the set of predicates is Erdős–Szekeres with Ramsey function .
In general, we consider an arbitrary finite set of semialgebraic predicates, meaning that each is a Boolean combination of polynomial equations and inequalities in some number of real variables. We define to be Erdős–Szekeres if for every there exists such that each -term sequence of real numbers has an -term subsequence such that at least one of the holds everywhere on , which means that holds for every choice of indices , . We write for the smallest 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 that is Erdős–Szekeres, there is a constant such that . Second, there is an algorithm that, given , decides whether it is Erdős–Szekeres; thus, -dimensional Erdős–Szekeres-style theorems can in principle be proved automatically.
We regard these results as a starting point in investigating analogous questions for -dimensional predicates, where instead of sequences of real numbers, we consider sequences of points in (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 (i.e., conjunctions of polynomial equations), as well as for a multipartite version of the problem with arbitrary semialgebraic predicates in .
"Erdős–Szekeres-type statements: Ramsey function and decidability in dimension ." Duke Math. J. 163 (12) 2243 - 2270, 15 September 2014. https://doi.org/10.1215/00127094-2785915