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Source. Theorem 16, pp. 574--575, with its proof, p. 575, of P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus, Euclidean Ramsey Theorems, III, Infinite and Finite Sets (Keszthely 1973), Colloq. Math. Soc. János Bolyai 10, North-Holland (1975), 559--583, as identified on the source card.

Statement

Theorem 16 (pp. 574--575). If $f$ is a proper two-coloring of $E^2$, there is a $(1,1,\sqrt3)$-triangle with the $120^\circ$ vertex colored oppositely from the other two. Thus all three colorings of an isosceles $120^\circ$ triangle occur in any proper two-coloring of $E^2$.

An isosceles triangle has three inequivalent colorings (p. 561). The other two are the monochromatic one, given for the $120^\circ$ triangle by Theorem 9 (i), and the one with a base vertex opposite, given by Theorem 12 (p. 573). The proof printed covers only the new coloring.

Proof pointer

P. 575. Like the proof of Theorem 14, with the triangular lattice: if the coloring is missing, each lattice $x+L$ with $L$ spanned by unit vectors at $60^\circ$ is constant along one of its three directions, and comparing with the rotated lattice spanned by $\tfrac17(5v+3u)$ and $\tfrac17(3v-5u)$ forces all pairs at distance $120$ to be like-colored.

Read depth. Claims checked: the statement was read on pp. 574--575; the proof was read for its structure only.

Bears on

• Problem 173: the monochromatic part of the conclusion, for the isosceles $120^\circ$ triangle, is already in Theorem 9; the theorem's own content is bichromatic and bears on the problem only through that case.