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Ramsey(3,3) = 6
Pegg Jr, Ed
  • none
  • The game of Sim, invented by Gustavus Simmons, matches Red against Blue on a hexagonal field of six dots. The players take turns drawing a line of their respective color between pairs of unconnected dots, losing if they make a triangle of their own color first. This Demonstration shows all the 32768 2-colorings of the hexagon. When a set of vertices makes a triangle, the vertices are circled. All of the colorings contain at least one triangle. The Ramsey problem R?(a,a) asks for the smallest n so that the complete graph Kn always contains a smaller monochromatic subgraph Ka, no matter how Kn is 2-colored. The graph that connects three points, K3, is a triangle. Since K5 can be 2-colored with no triangles (red star, blue pentagon), and since k6 always contains a triangle, the solution to the Ramsey problem R(3,3) is 6. The solution for R(4,4) is 18, with the 17-Paley graph and its inverse providing a 2-coloring for K17 without K4. The solution for R(5,5) is currently unknown, and it is predicted that the solution to R(6,6) will never be known
  • Componente Curricular::Ensino Fundamental::Séries Finais::Matemática
Issue Date: 
  • 2008
  • 8-Oct-2008
  • 8-Oct-2008
  • 8-Oct-2008
  • 23-Sep-2008
  • Combinatorics
  • Graph Theory
  • Unsolved Problems
  • Educação Básica::Ensino Fundamental Final::Matemática::Espaço e forma
This demonstration needs the "MathematicaPlayer.exe" to run. Found in
Demonstration freeware using Mathematica Player
Appears in Collections:MEC - Objetos Educacionais (BIOE) - OE

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