Discrete & Computational Geometry

, Volume 20, Issue 3, pp 375-388

First online:

Ramsey-Type Results for Geometric Graphs, II

  • Gy. KárolyiAffiliated withEötvös Loránd University, Múzeum krt. 6-8, 1088 Budapest, Hungary karolyi@cs.elte.hu
  • , J. PachAffiliated withCity College, C.U.N.Y., New York, NY 10031, USA
  • , G. TóthAffiliated withCourant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA pach@cims.nyu.edu, toth@cims.nyu.edu
  • , P. ValtrAffiliated withDepartment of Applied Mathematics, Charles University, Malostranské nám. 25, 11800 Praha 1, Czech Republic valtr@kam.ms.mff.cuni.cz

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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.