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Source. Theorem 12, Lemma 13 and their proofs, p. 573; Theorem 14 with its proof and Corollary 15, p. 574; 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

A two-coloring is proper when it is not a one-coloring. A triangle with three distinct sides has four inequivalent colorings: monochromatic, and the three in which one vertex differs from the other two (p. 561).

Theorem 14 (p. 574). If $f$ is a proper two-coloring of $E^2$ and $(a,b,c)$, $c^2=a^2+b^2$, is a right triangle with $b^2/a^2$ rational, then the $(a,b,c)$-triangle occurs in all four possible colorings.

Corollary 15 (p. 574). $R(a,b,c)$ holds whenever the ratio of two of the sides is $\sqrt2$. The paper derives it from the isosceles right triangle, which Theorem 14 covers.

Since a one-coloring makes every triangle monochromatic, Theorem 14 gives $R(a,b,c)$ for every right triangle with $b^2/a^2$ rational.

Theorem 12 (p. 573). For a proper two-coloring $f$ and every $a$, $b$ with $2a\ge b$, the isosceles $(a,a,b)$-triple satisfies $R_f(\bar a,a,b)$: some copy has the endpoints of one $a$-side alike and the third vertex opposite.

Lemma 13 (p. 573, from an argument of R. M. Robinson). Let $L=\{k+l\sqrt{-d}\mid k,l\in\mathbb Z\}$, $d>0$, $d\in\mathbb Q$, be a lattice in the complex plane. Then some rotation $L'$ of $L$ about $0$ by an angle that is not a multiple of $90^\circ$ has $L\cap L'$ a two-dimensional sublattice of $L$.

Proof pointer

P. 574. By Theorem 6 the monochromatic case follows from a bichromatic one, so it suffices to find the three bichromatic colorings. Normalize $a=1$ and take the rectangular lattice with sides $1$ and $b$. If one bichromatic coloring is missing, every lattice congruent to $2L$ has a coordinate direction along which each line is monochromatic. Lemma 13 supplies a rotated lattice $2L'$ meeting $2L$ in a sublattice; combining the monochromatic directions gives a monochromatic sublattice, and since $L$ was placed arbitrarily, all pairs at one fixed distance would be like-colored, contradicting properness. (The proof cites the lattice "of Lemma 12"; the lemma is numbered 13.)

Read depth. Claims checked: the statements were read clause by clause on pp. 573--574; the proof was read for its structure only.

Used by. Theorem 17, Corollary 20.

Bears on

• Problem 173: every right triangle with $b^2/a^2$ rational, and every triangle with two sides in ratio $\sqrt2$, has a monochromatic congruent copy in every two-coloring of the plane, so none is the exceptional triangle of any coloring.