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Statement and proof

In a red-blue coloring of the plane with no red unit-distance pair and no blue $\ell_5$, no equilateral triangle of side $3$ has all three vertices and its center red.

Suppose such a triangle has vertices $A,B,C$ and center $O$. Each vertex is at distance $\sqrt3$ from $O$. Choose

$$\theta=2\arcsin\!\left(\frac1{2\sqrt3}\right).$$

Rotate the three vertices through this angle about $O$, obtaining $A',B',C'$. The chord formula gives $|AA'|=|BB'|=|CC'|=2\sqrt3\sin(\theta/2)=1$. All three new vertices are therefore blue. Rotation preserves side lengths and the center, so $A',B',C'$ form a blue equilateral triangle of side $3$ with red center $O$. This contradicts Lemma 2.

Source and version corrections

Lemma 3, published p. 3, with Figure 1(b) on p. 2; Lemma 2.2 in arXiv v2. The published statement correctly uses side length $3$. The arXiv statement and a later sentence in its proof incorrectly use $\sqrt3$ for the triangle's side length. Both versions also call the initial vertices blue at the start of the proof; the intended initial vertices are red, as in the statement and figure. The proof above uses the published side length and the corrected initial color. Only elementary Euclidean rotation and chord length are needed beyond Lemma 2.

Bears on. #188.