On Geometric Graph Ramsey Numbers

  title={On Geometric Graph Ramsey Numbers},
  author={Gyula K{\'a}rolyi and Vera Rosta},
  journal={Graphs and Combinatorics},
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. 
2 Citations

On the geometric Ramsey numbers of trees

  • Pu Gao
  • Mathematics
    Discret. Math.
  • 2016

Ramsey-Type Problems for Geometric Graphs

We survey some results and collect a set of open problems related to graph Ramsey theory with geometric constraints.



Ramsey-Type Results for Geometric Graphs, I

Abstract. For any 2-coloring of the ${n \choose 2}$ segments determined by n points in general position in the plane, at least one of the color classes contains a non-self-intersecting spanning

Coloring arcs of convex sets

Ramsey-type results for geometric graphs. II

It is shown that for any 2{coloring of the n2 segments determined by n points in the plane, one of the color classes contains non-crossing cycles of lengths 3; 4; : : : ; bqn=2c, and it is proved that there is a non-Crossing path of length (n2=3), all of whose edges are of the same color.

Ramsey-type results for geometric graphs

Sharp estimates are given for the largest number of disjoint monochromatic triangles that can always be selected from the authors' segments, and there is a non-crossing path of length $\Omega(n^{2/3})$, all of whose edges are of the same color.

The Ramsey number for stripes

If G1,…,Gc are graphs without loops or multiple edges there is a smallest integer r(G1,…,Gc) such that if the edges of a complete graph Kn, with n ≧ r(G1,…,Gc), are painted arbitrarily with c colours

Tree-complete graph ramsey numbers

The ramsey number of any tree of order m and the complete graph of order n is 1 + (m − 1)(n − 1) where m is the number of trees and n is the total number of graphs.

Ramsey Theory Applications

There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical

Embedding a planar triangulation with vertices at specified points

Note: Professor Pach's number: [065] Reference DCG-ARTICLE-2008-010doi:10.2307/2323956 Record created on 2008-11-17, modified on 2017-05-12