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Coloring Points in the Plane and Elsewhere

There is a class of exciting problems that fall under the purview of what is called Euclidean Ramsey Theory. Points on a line, in a plane or space are assigned colors - are getting colored, so to speak. A configuration of points is said to be monochromatic if all the points in the configuration are of the same color. The theory clarifies the question of what kind of monochromatic configurations are there?

  1. Points in the plane are each colored with one of two colors: red or blue. Prove that, for a given distance d, there always exist two points of the same color at the distance d from each other. (Solution)

  2. Points in the plane are each colored with one of three colors: red, green, or blue. Prove that, for a given distance d, there always exist two points of the same color at the distance d from each other. (Solution)

  3. Points in the plane are each colored with one of two colors: red or blue. The set of distances between the blue points is blue and the set of distances between the red points is red. Prove that either one or the other contains all the positive reals. (Solution)

  4. Points on a straight line are colored in two colors. Prove that it is always possible to find three points of the same color with one being the midpoint of the other two. (Solution)

  5. Points in the plane are colored in two colors. Prove that it is always possible to find a monochromatic equilateral triangle, i.e., three points of the same color with all pairwise distances equal. (Solution)

  6. Is there a coloring of the plane with three colors such that any straight line is bichromatic, i.e. only contains points of two colors? (Solution)

  7. If each point of the plane is colored red or blue then some rectangle has its vertices all the same color. (Solution)

  8. Six points are given in the space such that the pairwise distances between them are all distinct. Consider the triangles with vertices at these points. Prove that the longest side of one of these triangles is at the same time the shortest side of another. (Solution

  9. The design obtained by cutting the plane with straight lines can be colored with just two colors so that no two regions that share a side are of the same color. (Solution)

References

  1. R. L. Graham, Euclidean Ramsey Theory, in Handbook of Discrete and Computational Mathematics, J. E. Goodman, J. O'Rourke (eds), Chapman & Hall/CRC, 2004
  2. R. B. J. T. Allenby, A. Slomson, How to Count: An Introduction to Combinatorics, CRC Press, 2011 (2nd edition)
  3. A. Soifer, Geometric Etudes in Combinatorial Mathematics, Springer, 2010 (2nd, expanded edition)
  1. Ramsey's Theorem
  2. Party Acquaintances
  3. Ramsey Number R(3, 3, 3)
  4. Ramsey Number R(4, 3)
  5. Ramsey Number R(5, 3)
  6. Ramsey Number R(4, 4)
  7. Geometric Application of Ramsey's Theory
  8. Coloring Points in the Plane and Elsewhere
  9. Two Colors - Two Points
  10. Three Colors - Two Points
  11. Two Colors - All Distances
  12. Two Colors on a Straight Line
  13. Two Colors - Three Points
  14. Three Colors - Bichromatic Lines
  15. Chromatic Number of the Plane
  16. Monochromatic Rectangle in a 2-coloring of the Plane
  17. Two Colors - Three Points on Circle
  18. Coloring a Graph
  19. No Equilateral Triangles, Please

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