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A classical and widely used lemma of Erdos and Szekeres asserts that for every n there exists N such that every N-term sequence a of real numbers contains an n-term increasing subsequence or an n-term nondecreasing subsequence; quantitatively, the smallest N with this property equals (n-1)^2+1.
A combinatorial problem in geometry
P. Erdős and G. Szekeres · 1935
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A decision method for elementary algebra and geometry
A. Tarski · 1951
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The classification of the order indiscernibles of real closed fields and other theories
D. A. Rosenthal · 1981
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On the number of halving planes
I. Bárány, Z. Füredi, and L. Lovász · 1990
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Ramsey Theory
R. L. Graham, B. L. Rothschild, and J. Spencer · 1990
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The colored Tverberg’s problem and complexes of injective functions
S. Vrećica and R. Živaljević · 1992
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Variations on the monotone subsequence theme of Erdős and Szekeres
M. J. Steele · 1995
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Real algebraic geometry
J. Bochnak, M. Coste, and M.-F. Roy · 1998
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An effective algorithm for quantifier elimination over algebraically closed fields using straight line programs
S. Puddu and J. Sabia · 1998
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The Erdős–Szekeres problem on points in convex position—a survey
W. Morris and V. Soltan · 2000
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Lectures on Discrete Geometry
J. Matoušek · 2002
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Algorithms in real algebraic geometry
S. Basu, R. Pollack, and M.-F. Roy · 2003
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Crossing patterns of semi-algebraic sets
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Betti numbers of semialgebraic sets defined by quantifier-free formulae
A. Gabrielov and N. Vorobjov · 2005
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T. Gerken · 2008
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Stabbing simplices by points and flats
B. Bukh, J. Matoušek, and G. Nivasch · 2010
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Lower bounds for weak epsilon-nets and stair-convexity
B. Bukh, J. Matoušek, and G. Nivasch · 2011
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One-sided epsilon-approximants
B. Bukh, P.-S. Loh, and G. Nivasch · 2012
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Higher-order Erdős–Szekeres theorems
M. Eliáš and J. Matoušek · 2012
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Overlap properties of geometric expanders
J. Fox, M. Gromov, V. Lafforgue, A. Naor, and J. Pach · 2012
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The empty hexagon theorem
C. M. Nicolás · 2007
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