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 2 of 4: Check triangles through

X

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In plain words

Check triangles through $X$

R_iB_j=R\Longleftrightarrow i\equiv j\pmod2

Detailed analysis

If $XY$ and $XZ$ have the same color, then $Y,Z$ lie in the same square. Their joining edge is either one of the uncolored diagonals or has the opposite color, so $XYZ$ is not monochromatic.