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Source. Theorem 6, printed pp. 344–345, physical pp. 4–5 of the published paper.

Let $S_3$ be an equilateral triangle of side $1$. Then

$$R(S_3,3,2)\ \text{is true}. \tag{1}$$

Proof

Color $\mathbb R^3$ red and blue. By Theorem 5, applied in any plane, there are same-colored points $A,C$ at distance $1$. After an isometry suppose they are red and

$$A=(-1/2,0,0),\qquad C=(1/2,0,0). \tag{2}$$

Their common unit-distance locus is the circle

$$\mathcal B=\{(0,y,z):y^2+z^2=3/4\}. \tag{3}$$

If any point of $\mathcal B$ is red, it completes a red unit equilateral triangle with $A,C$. Assume this does not happen. Then $\mathcal B$ is blue.

Write

$$R=\frac{\sqrt3}{2},\qquad \sin\alpha=\frac1{\sqrt3},\qquad a=R\cos\alpha=\frac1{\sqrt2}, \tag{4}$$

and in the $yz$-plane let

$$e_r(\theta)=(0,\cos\theta,\sin\theta),\qquad e_\theta(\theta)=(0,-\sin\theta,\cos\theta).$$

For every $\theta$, the two blue points

$$X_\theta=R e_r(\theta-\alpha), \qquad Y_\theta=R e_r(\theta+\alpha) \tag{5}$$

have distance $2R\sin\alpha=1$. Their midpoint is $M_\theta=a e_r(\theta)$, and the chord is parallel to $e_\theta(\theta)$. The common unit-distance locus of $X_\theta,Y_\theta$ is therefore the circle of radius $R$, centered at $M_\theta$, in the plane spanned by $(1,0,0)$ and $e_r(\theta)$. If one point of that circle were blue, it would form a blue unit equilateral triangle with $X_\theta,Y_\theta$. Consequently the entire surface

$$T(\theta,\phi) =R\sin\phi(1,0,0)+(a+R\cos\phi)e_r(\theta) \tag{6}$$

is red. This is the source's self-intersecting torus, now parametrized.

Choose $\phi$ so that

$$a+R\cos\phi=\frac1{\sqrt3}. \tag{7}$$

Such a $\phi$ exists because $a-R<0<1/\sqrt3<a+R$. At the three angles $0,2\pi/3,4\pi/3$, formula (6) has the same first coordinate and radial coordinate $1/\sqrt3$. Hence for distinct such angles

$$|T(\theta,\phi)-T(\theta+2\pi/3,\phi)|^2 =2\left(\frac1{\sqrt3}\right)^2 (1-\cos(2\pi/3))=1. \tag{8}$$

The three points are a red unit equilateral triangle, proving (1).

The paper had earlier observed that the corresponding planar two-color statement is false. That separate counterexample is not needed in this proof, and this page makes no present-day claim about other optimal dimensions.

Used by. Theorem 8.