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Color the edges (the complete graph on vertices) with three colors, red, green, and blue. When the Ramsey number , the graph will contain either a red , a green , or a blue , and any smaller complete graph can be colored to avoid such subgraphs.


The Ramsey number is known to be 17, so cannot be 3-colored without a monochromatic triangle.

This Demonstration shows that can be 3-colored to be triangle free, proving that . Every pair of circles is connected in exactly one of the three Clebsch graphs, which together form . Move the slider to permute the vertices.


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



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