# Ramsey(3,3) = 6

Initializing live version

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

[more]

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 asks for the smallest so that the complete graph always contains a smaller monochromatic subgraph , no matter how is 2-colored. The graph that connects three points, , is a triangle. Since can be 2-colored with no triangles (red star, blue pentagon), and since always contains a triangle, the solution to the Ramsey problem is 6. The solution for is 18, with the 17-Paley graph and its inverse providing a 2-coloring for without . The solution for is currently unknown, and it is predicted that the solution to will never be known.

[less]

Contributed by: Ed Pegg Jr (March 2011)
Open content licensed under CC BY-NC-SA

## Details

detailSectionParagraph

## Permanent Citation

Ed Pegg Jr "Ramsey(3,3) = 6"
http://demonstrations.wolfram.com/Ramsey336/
Wolfram Demonstrations Project
Published: March 7 2011

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send