Construct a 3-coloring of $K_{16}$ with no monochromatic triangle

Question

To prove that $R(3,3,3)>16$, I need to build an example of 3-coloring of $K_{16}$ which contains no monochromatic triangles.
Can anyone help me finding it?
Note : in this link I found that it has to do something with Petersen graph.

Answer 1

Take the 16 elements of the Galois field $\mathbb F_{16}$ as vertices and color the edge connecting $i$ to $j$ with the class of $i-j$ in $\mathbb F_{16}^\times/\mathbb F_{16}^{\times 3}$. Since there are 5 cubes in the field, there are 3 classes in this quotient.

Working this out with Mathematica gives me this coloring table for the edges:

      1   3   2   1   3   2   1   3   2   1   3   2   1   3   2
  1       1   1   3   1   3   2   3   1   2   2   2   3   3   2
  3   1       3   3   2   3   2   1   2   3   1   1   1   2   2
  2   1   3       2   2   1   2   1   3   1   2   3   3   3   1
  1   3   3   2       1   1   3   1   3   2   3   1   2   2   2
  3   1   2   2   1       3   3   2   3   2   1   2   3   1   1
  2   3   3   1   1   3       2   2   1   2   1   3   1   2   3
  1   2   2   2   3   3   2       1   1   3   1   3   2   3   1
  3   3   1   1   1   2   2   1       3   3   2   3   2   1   2
  2   1   2   3   3   3   1   1   3       2   2   1   2   1   3
  1   2   3   1   2   2   2   3   3   2       1   1   3   1   3
  3   2   1   2   3   1   1   1   2   2   1       3   3   2   3
  2   2   1   3   1   2   3   3   3   1   1   3       2   2   1
  1   3   1   3   2   3   1   2   2   2   3   3   2       1   1
  3   3   2   3   2   1   2   3   1   1   1   2   2   1       3
  2   2   2   1   2   1   3   1   2   3   3   3   1   1   3    

And a (useless) picture:

For reference, I computed the matrix as follows:

<< FiniteFields`
elements = GF[2, 4] /@ PowerList[GF[2, 4]];
allElements = Prepend[elements, 0];
cubes = Union[#^3 & /@ elements];
cubeClasses = Union@Table[Union[(# c) & /@ cubes], {c, elements}];
mat = Table[If[a =!= b, Position[cubeClasses, (a - b)][[1, 1]], 0], 
            {a, allElements}, {b, allElements}];