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Source. Theorem 17 with its proof, p. 576, 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 17 (p. 576). If $K$ is a right $(a,b,c)$-triangle with $a^2+b^2=c^2$, and the angle opposite the side of length $a$ is a rational multiple of $180^\circ$, then $R(K)$ is true.

Proof pointer

P. 576. Normalize $c=1$ and let $\beta$ be the angle opposite the leg $b$ in the paper's proof. Multiples of $90^\circ$ give a pair, which is trivially monochromatic. Theorem 7 extends this to $\beta=\tfrac{n}{2m+1}90^\circ$; Theorem 14 gives $\beta=\tfrac{2n+1}{2}90^\circ$, and the roulette method extends it to $\tfrac{2n+1}{2(2m+1)}90^\circ$; finally the bichromatic isosceles right triangle of Theorem 14 and the roulette method give $\tfrac{2n+1}{2(2m)}90^\circ$, which exhausts the rational multiples.

Read depth. Claims checked: the statement was read on p. 576; the proof was read for its structure only.

Bears on

• Problem 173: every right triangle with an acute angle a rational multiple of $180^\circ$ has a monochromatic congruent copy in every two-coloring of the plane, so none is the exceptional triangle of any coloring. Shader later proved this for every right triangle (see the problem page).