Problem source

Six points $A,B,C,D,E$ and $F$ lie in three dimensional space and are in general positions, that is, no three are collinear and no four lie on a plane. All possible line segments joining pairs of points are drawn and coloured either gold or silver. Prove that there is a triangle whose edges are entirely of one colour. 
{[}$\textbf{Hint}$: consider segments radiating from $A.${]}
Give a sketch showing that the result is false for five points in general positions.

Solution source

Consider the $5$ segements radiating from $A$. By the pigeonhole principle, at least $3$ of them must be the same colour (say gold and say reaching $B,C,D$). 

If any of the segments joining any of $B,C,D$ are gold then we have found a monochromatic gold triangle. But if none of them are gold, they are all silver, therefore $BCD$ is a monochromatic silver triangle.

\begin{center}
    \begin{tikzpicture}[scale=2]
        \def\r{2};
        \coordinate (A) at ({\r*cos(0)}, {\r*sin(0)});
        \coordinate (B) at ({\r*cos(1*72)}, {\r*sin(1*72)});
        \coordinate (C) at ({\r*cos(2*72)}, {\r*sin(2*72)});
        \coordinate (D) at ({\r*cos(3*72)}, {\r*sin(3*72)});
        \coordinate (E) at ({\r*cos(4*72)}, {\r*sin(4*72)});


        \filldraw (A) circle (1pt);
        \filldraw (B) circle (1pt);
        \filldraw (C) circle (1pt);
        \filldraw (D) circle (1pt);
        \filldraw (E) circle (1pt);

        \draw[ultra thick, yellow] (A) -- (B);
        \draw[ultra thick, yellow] (A) -- (C);
        \draw[ultra thick, lightgray] (A) -- (D);
        \draw[ultra thick, lightgray] (A) -- (E);

        \draw[ultra thick, lightgray] (B) -- (C);
        \draw[ultra thick, lightgray] (B) -- (D);
        \draw[ultra thick, yellow] (B) -- (E);

        \draw[ultra thick, yellow] (C) -- (D);
        \draw[ultra thick, lightgray] (C) -- (E);
        
        \draw[ultra thick, yellow] (D) -- (E);
    \end{tikzpicture}
\end{center}