Problem 3

Consider 9 points in space, no 4 coplanar. Each pair is joined by an edge colored red, blue, or left uncolored. Find the smallest

n

such that whenever exactly

n

edges are colored, there is a triangle whose three edges have the same color.

Step 3 of 4: Check triangles meeting one square twice

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In plain words

Check triangles meeting one square twice

i\equiv j\pmod2\Longrightarrow \chi(R_iR_j)=0

Detailed analysis

Consider $R_iR_jB_k$ . If $i\equiv j\pmod2$ , the edge $R_iR_j$ is an uncolored diagonal; otherwise $R_iB_k$ and $R_jB_k$ have opposite colors. The same argument applies to $B_iB_jR_k$ . Thus no such triangle is monochromatic.