## 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

#### References

• [1] N. Alon, J. Pach, R. Pinchasi, R. Radoičić, and M. Sharir, Crossing patterns of semi-algebraic sets, J. Combin. Theory Ser. A 111 (2005), 310–326.
• [2] I. Bárány, Z. Füredi, and L. Lovász, On the number of halving planes, Combinatorica 10 (1990), 175–183.
• [3] S. Basu, R. Pollack, and M.-F. Roy, Algorithms in Real Algebraic Geometry, Algorithms Comput. Math. 10, Springer, Berlin, 2003.
• [4] J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Ergeb. Math. Grenzgeb. (3) 36, Springer, Berlin, 1998.
• [5] B. Bukh, P.-S. Loh, and G. Nivasch, One-sided epsilon-approximants, in preparation.
• [6] B. Bukh, J. Matoušek, and G. Nivasch, Stabbing simplices by points and flats, Discrete Comput. Geom. 43 (2010), 321–338.
• [7] B. Bukh, J. Matoušek, and G. Nivasch, Lower bounds for weak epsilon-nets and stair-convexity, Israel J. Math. 182 (2011), 199–208.
• [8] D. Conlon, J. Fox, J. Pach, B. Sudakov, and A. Suk, Ramsey-type results for semi-algebraic relations, to appear in Trans. Amer. Math. Soc., preprint, arXiv:1301.0074v1 [math.CO].
• [9] M. Eliáš and J. Matoušek, Higher-order Erdős–Szekeres theorems, Adv. Math. 244 (2013), 1–15.
• [10] P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compos. Math. 2 (1935), 463–470.
• [11] J. Fox, M. Gromov, V. Lafforgue, A. Naor, and J. Pach, Overlap properties of geometric expanders, J. Reine Angew. Math. 671 (2012), 49–83.
• [12] A. Gabrielov and N. Vorobjov, Betti numbers of semialgebraic sets defined by quantifier-free formulae, Discrete Comput. Geom. 33 (2005), 395–401.
• [13] T. Gerken, Empty convex hexagons in planar point sets, Discrete Comput. Geom. 39 (2008), 239–272.
• [14] R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory, 2nd ed., Wiley, New York, 1990.
• [15] J. Matoušek, Lectures on Discrete Geometry, Grad. Texts in Math. 212, Springer, New York, 2002.
• [16] W. Morris and V. Soltan, The Erdős–Szekeres problem on points in convex position—a survey, Bull. Amer. Math. Soc. (N.S.) 37 (2000), 437–458.
• [17] C. M. Nicolás, The empty hexagon theorem, Discrete Comput. Geom. 38 (2007), 389–397.
• [18] S. Puddu and J. Sabia, An effective algorithm for quantifier elimination over algebraically closed fields using straight line programs, J. Pure Appl. Algebra 129 (1998), 173–200.
• [19] D. A. Rosenthal, The classification of the order indiscernibles of real closed fields and other theories, Ph.D. dissertation, University of Wisconsin, Madison, Wis., 1981.
• [20] M. J. Steele, “Variations on the monotone subsequence theme of Erdős and Szekeres” in Discrete Probability and Algorithms (Minneapolis, 1993), IMA Vol. Math. Appl. 72, Springer, New York, 1995, 111–131.
• [21] A. Tarski, A Decision Method for Elementary Algebra and Geometry, 2nd ed., Univ. of California Press, Berkeley, Calif., 1951.
• [22] R. T. Živaljević and S. T. Vrećica, The colored Tverberg’s problem and complexes of injective functions, J. Combin. Theory Ser. A 61 (1992), 309–318.