Erdős-Szekeres-type statements: Ramsey function and decidability in dimension 1. (English) Zbl 1301.05351

Summary: 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 \(\Phi=\{x_1<x_2,\;x_1\geq x_2\}\) is Erdős-Szekeres with Ramsey function \(\mathrm{ES}_\Phi (n)=(n-1)^2+1\). In general, we consider an arbitrary finite set \(\Phi=\{\Phi_1,\dots,\Phi_m\}\) 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 {\(\Phi\)} 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 \(i_1,i_2,\dots,i_k\), \(1\leq i_1<i_2<\cdots<i_k\leq n\). We write \(\mathrm{ES}_\Phi (n)\) 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 {\(\Phi\)} that is Erdős-Szekeres, there is a constant \(C\) such that \(\mathrm{ES}_\Phi (n)\leq 2^{2^{C_n}}\). Second, there is an algorithm that, given {\(\Phi\)}, 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 \(\mathbb R^d\).


05D10 Ramsey theory
52C45 Combinatorial complexity of geometric structures
Full Text: DOI arXiv Euclid


[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. · Zbl 1099.14048
[2] I. Bárány, Z. Füredi, and L. Lovász, On the number of halving planes , Combinatorica 10 (1990), 175-183. · Zbl 0718.52009
[3] S. Basu, R. Pollack, and M.-F. Roy, Algorithms in Real Algebraic Geometry , Algorithms Comput. Math. 10 , Springer, Berlin, 2003. · Zbl 1031.14028
[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. · Zbl 1387.05178
[6] B. Bukh, J. Matoušek, and G. Nivasch, Stabbing simplices by points and flats , Discrete Comput. Geom. 43 (2010), 321-338. · Zbl 1186.52001
[7] B. Bukh, J. Matoušek, and G. Nivasch, Lower bounds for weak epsilon-nets and stair-convexity , Israel J. Math. 182 (2011), 199-208. · Zbl 1222.68395
[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, [math.CO]. 1301.0074v1 · Zbl 1306.14027
[9] M. Eliáš and J. Matoušek, Higher-order Erdős-Szekeres theorems , Adv. Math. 244 (2013), 1-15. · Zbl 1283.05175
[10] P. Erdős and G. Szekeres, A combinatorial problem in geometry , Compos. Math. 2 (1935), 463-470. · Zbl 0012.27010
[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. · Zbl 1306.05171
[12] A. Gabrielov and N. Vorobjov, Betti numbers of semialgebraic sets defined by quantifier-free formulae , Discrete Comput. Geom. 33 (2005), 395-401. · Zbl 1073.14072
[13] T. Gerken, Empty convex hexagons in planar point sets , Discrete Comput. Geom. 39 (2008), 239-272. · Zbl 1184.52016
[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. · Zbl 0958.52018
[17] C. M. Nicolás, The empty hexagon theorem , Discrete Comput. Geom. 38 (2007), 389-397. · Zbl 1146.52010
[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. · Zbl 0929.68132
[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. · Zbl 0832.60012
[21] A. Tarski, A Decision Method for Elementary Algebra and Geometry , 2nd ed., Univ. of California Press, Berkeley, Calif., 1951. · Zbl 0044.25102
[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. · Zbl 0782.52003
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