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THE ERDŐS–SZEKERES PROBLEM FOR NON-CROSSING CONVEX SETS

Part of: Discrete geometry

Published online by Cambridge University Press:  14 May 2014

Michael Gene Dobbins ,
Andreas Holmsen  and
Alfredo Hubard
Michael Gene Dobbins
Affiliation:
GAIA, Postech, Pohang, South Korea email dobbins@postech.ac.kr
Andreas Holmsen
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon, South Korea email andreash@kaist.edu
Alfredo Hubard
Affiliation:
Département d’informatique, École Normale Supérior, Paris, France email hubard@di.ens.fr
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Abstract

We show an equivalence between a conjecture of Bisztriczky and Fejes Tóth about families of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and Tóth on the Erdős–Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk on the Erdős–Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erdős–Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erdős–Szekeres theorem of Pór and Valtr to families of non-crossing convex bodies.

MSC classification

Secondary: 52C10: Erdős problems and related topics of discrete geometry 52C30: Planar arrangements of lines and pseudolines
Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 463 - 484
Copyright
Copyright © University College London 2014 

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References

Bárány, I. and Károlyi, G., Problems and results around the Erdős–Szekeres convex polygon theorem. In Discrete and computational geometry (Tokyo, 2000) (Lecture Notes in Computer Science), Springer (Berlin, 2001), 91–105.Google Scholar
Bárány, I. and Valtr, P., A positive fraction Erdős–Szekeres theorem. Discrete Comput. Geom. 19 1998, 335–342.CrossRefGoogle Scholar
Bisztriczky, T. and Fejes Tóth, G., A generalization of the Erdős–Szekeres convex n-gon theorem. J. Reine Angew. Math. 395 1989, 167–170.Google Scholar
Bisztriczky, T. and Fejes Tóth, G., Nine convex sets determine a pentagon with convex sets as vertices. Geom. Dedicata 31 1989, 89–104.Google Scholar
Bisztriczky, T. and Fejes Tóth, G., Convexly independent sets. Combinatorica 10 1990, 195–202.CrossRefGoogle Scholar
Björner, A., Las Vergnas, M., Sturmfels, B., White, N. and Ziegler, G. M., Oriented matroids. In Encyclopedia of Mathematics and its Applications, 2nd edn, Cambridge University Press (1999).Google Scholar
Dhandapani, R., Goodman, J. E., Holmsen, A., Pollack, R. and Smorodinsky, S., Convexity in topological affine planes. Discrete Comput. Geom. 38 2007, 243–257.CrossRefGoogle Scholar
Dobbins, M. G., Holmsen, A. and Hubard, A., Regular systems of paths and families of convex sets in convex position. Trans. Amer. Math. Soc. (to appear).Google Scholar
Erdős, P. and Szekeres, G., A combinatorial problem in geometry. Compositio Math. 2 1935, 463–470.Google Scholar
Erdős, P. and Szekeres, G., On some extremum problems in elementary geometry. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3–4 1960/1961, 53–62.Google Scholar
Felsner, S. and Valtr, P., Coding and counting arrangements of pseudolines. Discrete Comput. Geom. 46 2011, 405–416.Google Scholar
Folkman, J. and Lawrence, J., Oriented matroids. J. Combin. Theory Ser. B 25 1978, 199–236.Google Scholar
Fox, J., Pach, J., Sudakov, B. and Suk, A., Erdős–Szekeres-type theorems for monotone paths and convex bodies. Proc. London Math. Soc. 105 2012, 953–982.CrossRefGoogle Scholar
Goodman, J. E., Proof of a conjecture of Burr, Grünbaum, and Sloane. Discrete Math. 32 1980, 27–35.CrossRefGoogle Scholar
Goodman, J. E., Pseudoline arrangements. In Handbook of Discrete and Computational Geometry (Discrete Mathematics and Its Applications), CRC (Boca Raton, FL, 1997), 83–109.Google Scholar
Goodman, J. E. and Pollack, R., On the combinatorial classification of nondegenerate configurations in the plane. J. Combin. Theory Ser. A 29 1980, 220–235.CrossRefGoogle Scholar
Goodman, J. E. and Pollack, R., A combinatorial perspective on some problems in geometry. In Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. I (Baton Rouge, LA, 1981), 383–394.Google Scholar
Goodman, J. E. and Pollack, R., Semispaces of configurations, cell complexes of arrangements. J. Combin. Theory Ser. A 37 1984, 257–293.CrossRefGoogle Scholar
Goodman, J. E. and Pollack, R., Upper bounds for configurations and polytopes in Rd. Discrete Comput. Geom. 1 1986, 219–227.Google Scholar
Goodman, J. E., Pollack, R., Wenger, R. and Zamfirescu, T., Arrangements and topological planes. Amer. Math. Monthly 101 1994, 866–878.Google Scholar
Groemer, H., Geometric applications of Fourier series and spherical harmonics. In Encyclopedia of Mathematics and its Applications, Cambridge University Press (1996).Google Scholar
Grünbaum, B., Arrangements and spreads, American Mathematical Society (Providence, RI, 1972).CrossRefGoogle Scholar
Habert, L. and Pocchiola, M., LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies. Discrete Comput. Geom. 50 2013, 552–648.Google Scholar
Hubard, A., Montejano, L., Mora, E. and Suk, A., Order types of convex bodies. Order 28 2011, 121–130.Google Scholar
Knuth, D. E., Axioms and Hulls (Lecture Notes in Computer Science 606), Springer (Berlin, 1992).Google Scholar
Morris, W. and Soltan, V., The Erdős–Szekeres problem on points in convex position—a survey. Bull. Amer. Math. Soc. (N.S.) 37 2000, 437–458.Google Scholar
Pach, J. and Solymosi, J., Canonical theorems for convex sets. Discrete Comput. Geom. 19 1998, 427–435.CrossRefGoogle Scholar
Pach, J. and Tóth, G., A generalization of the Erdős–Szekeres theorem to disjoint convex sets. Discrete Comput. Geom. 19 1998, 437–445.Google Scholar
Pach, J. and Tóth, G., Erdős–Szekeres-type theorems for segments and noncrossing convex sets. Geom. Dedicata 81 2000, 1–12.CrossRefGoogle Scholar
Pór, A. and Valtr, P., The partitioned version of the Erdős–Szekeres theorem. Discrete Comput. Geom. 28 2002, 625–637.Google Scholar
Pór, A. and Valtr, P., On the positive fraction Erdős–Szekeres theorem for convex sets. European J. Combin. 27 2006, 1199–1205.Google Scholar
Ringel, G., Über Geraden in allgemeiner Lage. Elem. Math. 12 1957, 75–82.Google Scholar
Szekeres, G. and Peters, L., Computer solution to the 17-point Erdős–Szekeres problem. ANZIAM J. 48 2006, 151–164.CrossRefGoogle Scholar
Tóth, G., Finding convex sets in convex position. Combinatorica 20 2000, 589–596.Google Scholar
Tóth, G. and Valtr, P., The Erdős–Szekeres theorem: upper bounds and related results. In Combinatorial and Computational Geometry (Mathematical Sciences Research Institute Publications 52), Cambridge University Press (Cambridge, 2005), 557–568.Google Scholar