> [!NOTE] Lemma > If every element of $\mathbb{R}^3$ is coloured red, green, or blue then there exists one colour such that, any positive real number represents the distance between two points of this colour (that is, one colour attains all distances). ###### Proof BWOC suppose no colour attains all distances, that is there exists $r,g,b$ such that every point at a distance $r$ from any red point is either green or blue and so on. WLOG suppose $r\geq g \geq b.$ Consider the sphere $S_{r}$ of radius $r$ centred at some red point $P_{r}$. Then its surface has green and blue points only. Since $g,b\leq r,$ the surface of the sphere contains both green and blue points (it can't be monochromatic since the points on the surface of the sphere of radius $r$ attains all distances between $0$ and $2r$). Choose a green point $P_{g}$ on the surface of $S_{r}$. Consider the sphere $S_{g}$ of radius $g\leq r$ centred at $P_{g},$ which intersects $S_{r}$ in a circle $C_{g}$ that is only blue. Now pick a point $P_{b}$ on $C_{g}$, and draw the sphere $S_{b}$ around $P_b$, which intersects circle $C_{g}$ in at least one point since $b\leq g.$ It follows that the distance between these two points is $b$, but you have no more colours to use, as $C_g$ is entirely blue, contradicting the initial assumption. Note that $C_{g}$ has radius $\frac{\sqrt{ 3 }}{2} \; g\leq \rho<g$ and it attains the lower bound when $g=r.$ Thus we can't make $\rho$ so small that $b$ exceeds $2\rho$ since $2\rho \geq \sqrt{ 3 }g >g \geq b.$ # Applications Takeaways: WLOG, make use of ordering. Allows you to think of just one possibility. Evaluate accuracy of your diagram. Questions: Expected distance between two random points on sphere surface? # References (German Mathematical Olympiad, 1985) Gelca, Andreescu