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On Geometric Graph Ramsey Numbers

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Abstract

For any two-colouring of the segments determined by 3n − 3 points in general position in the plane, either the first colour class contains a triangle, or there is a noncrossing cycle of length n in the second colour class, and this result is tight. We also give a series of more general estimates on off-diagonal geometric graph Ramsey numbers in the same spirit. Finally we investigate the existence of large noncrossing monochromatic matchings in multicoloured geometric graphs.

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References

  1. Araujo, G., Dumitrescu, A., Hurtado, F., Noy, M., Urrutia, J.: On the chromatic number of some geometric Kneser graphs. Comput. Geom. Theory Appl. 32, 59–69 (2005)

    Google Scholar 

  2. Bialostocki, A., Harborth, H.: Ramsey colorings for diagonals of convex polygons. Abh. Braunschweig. Wiss. Ges. 47, 159–163 (1996)

    Google Scholar 

  3. Chvátal, V.: Tree—complete graph Ramsey numbers. J. Graph Theory 1, 93 (1977)

    Google Scholar 

  4. Cockayne, E.J., Lorimer, P.J.: The Ramsey number for stripes. J. Austral. Math. Soc. A 19, 252–256 (1975)

    Google Scholar 

  5. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)

    Google Scholar 

  6. Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory, 2nd edn. Wiley, New York (1990)

  7. Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points (solution to problem E3341). Amer. Math. Monthly 98, 165–166 (1991)

    Google Scholar 

  8. Harborth, H., Lefmann, H.: Coloring arcs of convex sets. Discrete Math. 220, 107–117 (2000)

    Google Scholar 

  9. Károlyi, Gy., Pach, J., Tóth, G.: Ramsey-type results for geometric graphs. I. Discrete Comput. Geom. 18, 247–255 (1997)

    Google Scholar 

  10. Károlyi, Gy., Pach, J., Tóth, G., Valtr, P.: Ramsey-type results for geometric graphs. II. Discrete Comput. Geom. 20, 375–388 (1998)

  11. Károlyi, Gy., Rosta, V.: Generalized and geometric Ramsey numbers. Theor. Comput. Sci. 263, 87–98 (2001)

    Google Scholar 

  12. Nešetřil, J., Rödl, V. (Eds.): Mathematics of Ramsey Theory. Algorithms and Combinatorics, 5. Springer, Heidelberg (1990)

  13. Ramsey, F.P.: On a problem of formal logic. Proc. London Math. Soc. 30, 264–286 (1930)

    Google Scholar 

  14. Rosta, V.: Ramsey theory applications. Electron. J. Combin. 11, Dynamic Survey DS13 (electronic), 43 pages (2004)

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Authors and Affiliations

  1. Department of Algebra and Number Theory, Eötvös University, Pázmány P. sétány 1/C, Budapest, H–1117, Hungary

    Gyula Károlyi

  2. Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13–15, Budapest, H–1053, Hungary

    Vera Rosta

Authors
  1. Gyula Károlyi
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  2. Vera Rosta
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Correspondence to Gyula Károlyi.

Additional information

Research partially supported by Hungarian Scientific Research Grants OTKA T043631 and NK67867.

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Károlyi, G., Rosta, V. On Geometric Graph Ramsey Numbers. Graphs and Combinatorics 25, 351–363 (2009). https://doi.org/10.1007/s00373-009-0847-7

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  • DOI: https://doi.org/10.1007/s00373-009-0847-7

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