Abstract
A crossing family is a collection of pairwise crossing segments, this concept was introduced by Aronov et al. [4]. They proved that any set of n points (in general position) in the plain contains a crossing family of size \({\sqrt{n/12}}\). In this paper we present a generalization of the concept and give several results regarding this generalization.
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References
O. Aichholzer, Enumerating order types for small point sets with applications, www.ist.tugraz.at/staff/aichholzer/research/rp/triangulations/ordertypes/, (2006), accessed August 14, 2018.
Aichholzer O., Aurenhammer F., Krasser H.: Enumerating order types for small point sets with applications. Order, 19, 265–281 (2002)
J. L. Álvarez, J. Cravioto, and J. Urrutia, Crossing families and self crossing hamiltonian cycles, in: Abstracts of the XVI Spanish Meeting on Computational Geometry (Barcelona, July 1–3, 2015), 13–16.
Aronov B., Erdős P., Goddard W., Kleitman D.J., Klugerman M., Pach J., Schulman L.J.: Crossing families. Combinatorica, 14, 127–134 (1994)
I. Bárány and P. Valtr, A positive fraction Erdős–Szekeres theorem, Discrete Comput. Geom., 19 (1998) 335–342 (special issue, dedicated to the memory of Paul Erdős).
Ceder J.G.: Generalized sixpartite problems. Bol. Soc. Mat. Mexicana (2), 9, 28–32 (1964)
Reinhard Diestel, Graph Theory (Chapter Planar Graphs, p. 92), Springer (2005).
Erdős P.: On sets of distances of n points. Amer. Math. Monthly, 53, 248–250 (1946)
Erdős P., Szekeres G.: A combinatorial problem in geometry. Compositio Math., 2, 463–470 (1935)
C. Huemer, D. Lara, and C. Rubio-Montiel, Coloring decompositions of complete geometric graphs, arXiv:1610.01676v4 [math.CO].
D. Lara and C Rubio-Montiel, On crossing families of complete geometric graphs, arXiv: 1805.09888 [math.CO].
D. Lara and C Rubio-Montiel, On crossing families of complete geometric graphs, http://delta.cs.cinvestav.mx/~dlara/crossing_families.pdf.
Megiddo N.: Partiotioning with two lines in the plane. J. Algorithms, 6, 430–433 (1985)
Nielsen M.J., Sabo D.E.: Transverse families of matchings in the plane. Ars Combin., 55, 193–199 (2000)
Nielsen M.J, Webb W.: On some numbers related to the Erdős–Szekeres theorem. Open J. Discrete Math., 3, 167–173 (2013)
J. Pach and J. Solymosi, Halving lines and perfect cross-matchings, in: Advances in Discrete and Computational Geometry (South Hadley, MA, 1996), Contemp. Math., vol. 223, Amer. Math. Soc. (Providence, RI, 1999), pp. 245–249.
Suk A.: On the Erdős–Szekeres convex polygon problem. J. Amer. Math. Soc., 30, 1047–1053 (2017)
Szekeres G., Peters L.: Computer solution to the 17-point Erdős–Szekeres problem. ANZIAM J., 48, 151–164 (2006)
G. Tóth and P. Valtr, The Erdős–Szekeres theorem: upper bounds and related results, in: Combinatorial and Computational Geometry, Math. Sci. Res. Inst. Publ., vol. 52, Cambridge Univ. Press (Cambridge, 2005), pp. 557–568.
P. Valtr, On mutually avoiding sets, in: The Mathematics of Paul Erdős, II, Algorithms Combin., vol. 14, Springer (Berlin, 1997), pp. 324–332.
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The authors thank the referee for helpful advice on an earlier draft of the paper.
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Lara, D., Rubio-Montiel, C. On crossing families of complete geometric graphs. Acta Math. Hungar. 157, 301–311 (2019). https://doi.org/10.1007/s10474-018-0880-1
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DOI: https://doi.org/10.1007/s10474-018-0880-1