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Source: published paper, printed p. 219, the unnumbered unit-sphere observation.

Statement

If $K$ is any subset of a sphere of radius one in $\mathbb R^n$, then $\mathbb E^n\to(\ell_2,K)$. The set $K$ need not be finite.

Full proof

Assume there is no red unit-distance pair. If there are no red points, the whole space is blue and contains $K$. Otherwise choose a red point $p$. Every point of the sphere of radius one about $p$ is blue. Translate the sphere containing $K$ to that sphere; the corresponding translate of $K$ is blue. This proves the assertion.

This observation explains why cardinality alone cannot upper-bound the size of every configuration forced against a red unit pair. The separation and diameter conditions in the main theorem have substantive roles.

Related proof pages. theorem 1 2.

Bears on. Problem 188, Problem 214.