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I tried to solve this problem, but I'm not sure I got it right. The question proposes a hint:

Let G be a clique with 6 vertices which it's edges are colored with 3 different colors, then there's a monochromatic triangle in G.

This is my suggestion:

Let $G=(V,E)$ and $v_1\in V$. $G$ is a clique with 17 vertices, hence there are 16 edges from $v_1$. There are 3 colors, hence there are at least $\lceil16 / 3\rceil=6$ edges with the same color from $v_1$, say Blue.

Now consider the vertices $\{v_2,...,v_7\}$, neighbors of $v_1$. These are 6 vertices in this group and they perform a clique with 6 vertices. If the edges between them are colored with only 2 colors, we're done, since every clique with 6 vertices and 2 colors of edges have a monochromatic triangle. Otherwise there are 3 colors, and we can conclude that there's an blue edges, say $(v_i,v_j)$ for $i\ne j \in {2,...,7}$. Notice we got the blue triangle: $v_1 - v_i - v_j$.

Is there any problem with this proof?

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  • $\begingroup$ If you have a clique of six which is coloured with TWO colours, you get a monochromatic triangle. Whether a triangle is monochromatic or not depends on specifics of the colouring - you haven't justified that the $180$ triangles are evenly split between different colourings. In fact you could colour the edges from clique A to clique B with colour $1$, from B to C with colour two and from C to A with colour $3$. Then all your 180 triangles are identically coloured with one side of each colour, and none is monochromatic. $\endgroup$ – Mark Bennet Sep 30 '18 at 16:15
  • $\begingroup$ Thanks, I'll edit now and suggest another proof that I came up with. $\endgroup$ – E_K Sep 30 '18 at 16:42
  • $\begingroup$ I think the proof is OK now, try to generalize it, to prove that if for every coloring of a clique with $f(r-1)$ vertices with $r-1$ color there is a monochromatic triangle, then for every coloring of a clique with $r(f(k-1)-1)+2$ vertices with $r$ colors there is a monochromatic triangle :D. $\endgroup$ – Marcelo Campos Sep 30 '18 at 21:36
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    $\begingroup$ @Marcelo Campos I'm a little confused. What is k? Do you mean $r\cdot (f(k-1)-1)+2$ vertices? $\endgroup$ – E_K Oct 1 '18 at 20:58
  • $\begingroup$ Yes, exactly, sorry for the confusing notation. $\endgroup$ – Marcelo Campos Oct 11 '18 at 19:38
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I think your proof is fine; however it is useful to know that this is a small example of what is a larger class of problems. These are numbers called Ramsey numbers, and here you prove that $R(3,3,3)$ is $17$. You might want to check out Chromatic Triangles on a $K_{17}$ graph and this. I personally find them fascinating and they're a somewhat celebrated topic.

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