Abstract
According to the Erdős–Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erdős–Szekeres theorem.
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References
Avis, D., Hosono, K., Urabe, M.: On the existence of a point subset with a specified number of interior points. Discrete Math. 241(1), 33–40 (2001)
Bárány, I., Károlyi, G.: Problems and results around the Erdős–Szekeres convex polygon theorem. In: Akiyama, J., Kano, M., Urabe, M. (eds.) Discrete and Computational Geometry (Tokyo, 2000). Lecture Notes in Computer Science, vol. 2098, pp. 91–105. Springer, Berlin (2001)
Bárány, I., Pach, J.: Homogeneous selections from hyperplanes. J. Comb. Theory Ser. B 104, 81–87 (2014)
Bárány, I., Valtr, P.: A positive fraction Erdős–Szekeres theorem. Discrete Comput. Geom. 19, 335–342 (1998)
Christ, T.: Discrete descriptions of geometric objects. Ph.D. Thesis, ETH Zurich Institute for Theoretical Computer Science. http://e-collection.library.ethz.ch/view/eth:5242 (2011)
Chung, F.R.K., Graham, R.L.: Forced convex \(n\)-gons in the plane. Discrete Comput. Geom. 19(3), 367–371 (1998)
Erdős, P.: Some applications of graph theory and combinatorial methods to number theory and geometry. In: Lovász, L., Sós, V.T. (eds.) Algebraic Methods in Graph Theory, Vol. I, II (Szeged, 1978). Colloquia Mathematica Societatis János Bolyai, vol. 25, pp. 137–148. North-Holland, Amsterdam (1981)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)
Erdős, P., Szekeres, G.: On some extremum problems in elementary geometry. Ann. Univ. Sci. Bp. Eötvös Sect. Math. 3–4, 53–62 (1960)
Felsner, S., Kriegel, K.: Triangles in Euclidean arrangements. Discrete Comput. Geom. 22(3), 429–438 (1999)
Fox, J., Gromov, M., Lafforgue, V., Naor, A., Pach, J.: Overlap properties of geometric expanders. J. Reine Angew. Math. 671, 49–83 (2012)
Füredi, Z., Palásti, I.: Arrangements of lines with a large number of triangles. Proc. Am. Math. Soc. 92(4), 561–566 (1984)
Gerken, T.: Empty convex hexagons in planar point sets. Discrete Comput. Geom. 39(2), 239–272 (2008)
Grünbaum, B.: Arrangements and Spreads, vol. 10. American Mathematical Society, Providence, RI (1972)
Harborth, H.: Konvexe Fünfecke in ebenen Punktmengen. Elem. Math. 33(5), 116–118 (1978)
Harborth, H., Möller, M.: The Esther Klein problem in the projective plane. J. Combin. Math. Combin. Comput. 15, 171–179 (1994)
Horton, J.D.: Sets with no empty convex \(7\)-gons. Can. Math. Bull. 26(4), 482–484 (1983)
Kleitman, D., Pachter, L.: Finding convex sets among points in the plane. Discrete Comput. Geom. 19(3), 405–410 (1998). Dedicated to the memory of Paul Erdős
Leanos, J., Lomeli, M., Merino, C., Salazar, G., Urrutia, J.: Simple Euclidean arrangements with no (\(\ge \) \(5\))-gons. Discrete Comput. Geom. 38(3), 595–603 (2007)
Levi, F.: Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade. Ber. Math. Phys. Kl. Sächs. Akad. Wiss 78, 256–267 (1926)
Morris, W., Soltan, V.: The Erdős–Szekeres problem on points in convex position—a survey. Bull. Am. Math. Soc. New Ser. 37(4), 437–458 (2000)
Nicolás, C.M.: The empty hexagon theorem. Discrete Comput. Geom. 38(2), 389–397 (2007)
Pach, J., Solymosi, J.: Canonical theorems for convex sets. Discrete Comput. Geom. 19(3), 427–435 (1998). Dedicated to the memory of Paul Erdős
Pór, A., Valtr, P.: The partitioned version of the Erdős–Szekeres theorem. Discrete Comput. Geom. 28(4), 625–637 (2002)
Szekeres, G., Peters, L.: Computer solution to the 17-point Erdős–Szekeres problem. ANZIAM J. 48(2), 151–164 (2006)
Tóth, G., Valtr, P.: Note on the Erdős–Szekeres theorem. Discrete Comput. Geom. 19(3), 457–459 (1998)
Tóth, G., Valtr, P.: The Erdős–Szekeres theorem: upper bounds and related results. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry. Mathematical Sciences Research Institute Publications, vol. 52, pp. 557–568. Cambridge University Press, Cambridge (2005)
Wei, X., Ding, R.: More on an Erdős–Szekeres-type problem for interior points. Discrete Comput. Geom. 42(4), 640–653 (2009)
Acknowledgments
Research of the first and second authors was partially supported by ERC Advanced Research Grant No. 267165 (DISCONV), and research of the first and third authors by Hungarian Science Foundation Grant OTKA K 83767 and K 111827. Research of the third author was also supported by Hungarian Science Foundation Grant OTKA NN 102029 under the EuroGIGA programs ComPoSe and GraDR. We would like to thank Miguel Raggi who offered to write the program that determined the value of \({\text {ES}}_l(5)\).
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Bárány, I., Roldán-Pensado, E. & Tóth, G. Erdős–Szekeres Theorem for Lines. Discrete Comput Geom 54, 669–685 (2015). https://doi.org/10.1007/s00454-015-9705-y
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DOI: https://doi.org/10.1007/s00454-015-9705-y