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.