---

Source. Theorem 6, p. 566, with Figure 2 and the proof, pp. 566--568, 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 6 (p. 566). Let $f$ be a two-coloring of $E^2$ and $n$ a positive integer. Let $a^2+b^2=c^2$ and let $K$ be a triple with sides $a,b,c$. Let $K'$ have sides $a/(2n+1),b,c'$ with $a^2/(2n+1)^2+b^2=(c')^2$, and $K''$ sides $a/(2n),b,c''$ with $a^2/(2n)^2+b^2=(c'')^2$. Then:

• if $R_f(K)$ holds, so does $R_f(K')$;
• if some $K^*$ congruent to $K$ has its two points at distance $b$ of one color and its third point of the other color, then $R_f(K'')$ holds;
• if such a $K^*$ exists, then there is a triple with sides $a/n,b,d$, $(a/n)^2+b^2=d^2$, whose two points at distance $b$ have the same color and whose third point has the opposite color.

The paper calls this the "ladder method" (p. 566).

Proof pointer

Pp. 566--568. Start from like-colored points $x,y$ at distance $b$. If the right triangle with legs $a'$ and $b$ is never monochromatic, the points at distance $a'$ from $x$ and $y$ perpendicular to $xy$ take the other color, and iterating gives a ladder (Figure 2, p. 567) whose rungs alternate in color; with $a'=a/(2n+1)$ or $a'=a/(2n)$ the third vertex of the given triangle sits on a rung of the wrong color. The third statement uses a ladder of constant color.

Read depth. Claims checked: the statement was read clause by clause on p. 566; the proof was read for its structure only.

Used by. Theorem 14, and the lists of right triangles in $T_f$ on pp. 577--578.

Bears on

• Problem 173: a conditional tool. It transfers the existence of monochromatic right triangles from one shape to others within a fixed coloring and, through Theorem 14, enters the proof of $R(K)$ for right triangles with $b^2/a^2$ rational. It decides no triangle by itself.