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Source. Theorem 7, p. 568, with its proof and Figure 3, pp. 568--569, 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 7 (p. 568). Let $f$ be a two-coloring of $E^2$. Let $K_\alpha$ be a triple with angles $90^\circ,\alpha,90^\circ-\alpha$, where $\alpha=0$ is allowed (the triple then degenerates to a pair), and let $K_\beta$ be a triple with angles $90^\circ,\beta,90^\circ-\beta$ and the same hypotenuse length as $K_\alpha$.

• If $R_f(K_\alpha)$ holds, then $R_f(K_\beta)$ holds when $(2m+1)\beta=\alpha+n\cdot180^\circ$ for some integers $m\ge0$, $n\ge0$.
• If some $K'_\alpha\cong K_\alpha$ has its two hypotenuse vertices of one color and its third vertex of the other, then $R_f(K_\beta)$ holds if $2m\beta=\alpha+n\cdot180^\circ$.
• If such a $K'_\alpha$ exists, then there is a triangle $K'_\beta$, with the same hypotenuse length, whose $90^\circ$ vertex is colored opposite to the other two, if $m\beta=\alpha+n\cdot180^\circ$.

The second and third statements print no range for $m$, $n$. The paper calls this the "roulette method" (p. 568).

Proof pointer

Pp. 568--569. On the circle with diameter $xy$, $x,y$ like-colored at the hypotenuse distance $c$, if $R_f(K_\beta)$ fails the points obtained by turning through the angle $\beta$ at $x$ and $y$ alternate in color (Figure 3, p. 569); when the stated relation holds, the third vertex of the given copy of $K_\alpha$ lands on one of them with the wrong color. The other statements run the same way.

Read depth. Claims checked: the statement was read clause by clause on p. 568; the proof was read for its structure only.

Used by. Theorem 17 and Theorems 18 and 19 (p. 576).

Bears on

• Problem 173: a conditional tool that moves monochromatic right triangles between angles within a fixed coloring; through Theorem 17 it gives $R(K)$ for right triangles with an angle a rational multiple of $180^\circ$. It decides no triangle by itself.