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Source. Theorem 5 and Figure 1, printed p. 344, physical p. 4 of the published paper.

Let $P_d$ be a pair of points at distance $d>0$. Then

$$R(P_d,2,7)\ \text{is false}, \qquad R(P_d,2,3)\ \text{is true}. \tag{1}$$

Here $R(K,n,r)$ means that every $r$-coloring of $\mathbb R^n$ contains a monochromatic congruent copy of $K$.

The three-color assertion

The exact coordinate construction and independence-number calculation for the source's seven-point spindle are given in the canonical seven-point spindle proof. For every $d>0$, that result supplies a seven-point planar set $W_d$ such that a subset of $W_d$ containing no pair at distance $d$ has at most two points.

Restrict any three-coloring of the plane to $W_d$. If it had no monochromatic pair at distance $d$, each of its three color classes would contain at most two points of $W_d$. The three classes could then contain at most six of the seven points, a contradiction. This proves the positive assertion in (1).

The seven-color assertion

The paper does not print the plane coloring proving the first assertion of (1); it refers to its references [4] and [2]. That assertion is retained as an exact external result, not as a proof reconstructed on this page.

Used by. Theorem 6 and Theorem 9.