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Discrete & Computational Geometry

, Volume 20, Issue 3, pp 375–388 | Cite as

Ramsey-Type Results for Geometric Graphs, II

  • Gy. Károlyi
  • J. Pach
  • G. Tóth
  • P. Valtr

Abstract.

We show that for any two-coloring of the \({n \choose 2}\) segments determined by n points in the plane, one of the color classes contains noncrossing cycles of lengths \(3,4,\ldots,\lfloor\sqrt{n/2}\rfloor\) . This result is tight up to a multiplicative constant. Under the same assumptions, we also prove that there is a noncrossing path of length Ω(n 2/3 ) , all of whose edges are of the same color. In the special case when the n points are in convex position, we find longer monochromatic noncrossing paths, of length \(\lfloor({n+1})/{2}\rfloor\) . This bound cannot be improved. We also discuss some related problems and generalizations. In particular, we give sharp estimates for the largest number of disjoint monochromatic triangles that can always be selected from our segments.

Keywords

Related Problem Multiplicative Constant Sharp Estimate Color Classis Geometric Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • Gy. Károlyi
    • 1
  • J. Pach
    • 2
  • G. Tóth
    • 3
  • P. Valtr
    • 4
  1. 1.Eötvös Loránd University, Múzeum krt. 6-8, 1088 Budapest, Hungary karolyi@cs.elte.huHU
  2. 2.City College, C.U.N.Y., New York, NY 10031, USAUS
  3. 3.Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA pach@cims.nyu.edu, toth@cims.nyu.eduUS
  4. 4.Department of Applied Mathematics, Charles University, Malostranské nám. 25, 11800 Praha 1, Czech Republic valtr@kam.ms.mff.cuni.czCZ

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