Chromatic Triangles on a $K_{17}$ graph

Yes. Choose one vertex. It has sixteen edges going out, so six of some color, say yellow. Now consider the $K_6$ composed of those six vertices. If it has no yellow edges, it has two monochromatic triangles and we are done. If it has two yellow edges, we have two monochromatic triangles and are again done. If it has only one yellow edge we have one monochromatic triangle. Now choose some vertex not involved in this argument-we have only used seven, so there are ten more. It must invoke at least one monochromatic triangle by the same argument.

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Graph Theory

Ramsey Theory

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