Duke Mathematical Journal

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

Boris Bukh and Jiří Matoušek

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A classical and widely used lemma of Erdős and Szekeres asserts that for every n there exists N such that every N-term sequence a̲ of real numbers contains an n-term increasing subsequence or an n-term nonincreasing subsequence; quantitatively, the smallest N with this property equals (n1)2+1. In the setting of the present paper, we express this lemma by saying that the set of predicates Φ={x1<x2,x1x2} is Erdős–Szekeres with Ramsey function ESΦ(n)=(n1)2+1.

In general, we consider an arbitrary finite set Φ={Φ1,,Φm} of semialgebraic predicates, meaning that each Φj=Φj(x1,,xk) is a Boolean combination of polynomial equations and inequalities in some number k of real variables. We define Φ to be Erdős–Szekeres if for every n there exists N such that each N-term sequence a̲ of real numbers has an n-term subsequence b̲ such that at least one of the Φj holds everywhere on b̲, which means that Φj(bi1,,bik) holds for every choice of indices i1,i2,,ik, 1i1<i2<<ikn. We write ESΦ(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 Φ that is Erdős–Szekeres, there is a constant C such that ESΦ(n)22Cn. Second, there is an algorithm that, given Φ, 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 Rd (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 Rd (i.e., conjunctions of polynomial equations), as well as for a multipartite version of the problem with arbitrary semialgebraic predicates in Rd.

Article information

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

First available in Project Euclid: 15 September 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05D10: Ramsey theory [See also 05C55]
Secondary: 52C45: Combinatorial complexity of geometric structures [See also 68U05]


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

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