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Source. Conjectures 2 and 3 and the passage between them, p. 560; the reformulations on pp. 564 and 565; 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

Conjecture 2 (p. 560), as posed: "If $E^2$ is 2-colored so that there is no equilateral triangle of side $d$, then there is a monochromatic equilateral triangle of side $d'$, for $d'\ne d$." The hypothesis is read with "monochromatic" understood, as the surrounding text and the reformulation on p. 565 show: the paper restates Conjecture 2 as saying that $T_f=\{(a,a,a)\}$ for some $a>0$, or $T_f=\emptyset$, so the conclusion is for every $d'\ne d$.

Conjecture 3 (p. 560), as posed: "If $K$ is a triangle which is not equilateral, then $R(K)$ is true."

The notation $R(K)$, $R_f(K)$ and $T_f$ is that of Theorem 1. The paper derives Conjecture 3 from Conjecture 2 through Theorem 1 and states that the two are equivalent in view of Theorem 1 (p. 560); it restates Conjecture 3 as $T_f\subset\{(a,a,a)\mid a>0\}$ for every two-coloring $f$ (p. 564). It notes (p. 582) that it would suffice to prove Conjecture 3 for the non-equilateral isosceles triangles, and that it has no $(a,a,b)$-triangle with $R(a,a,b)$ and $a/b$ transcendental.

Status in the paper

Posed, not proved. The paper proves $R(K)$ for several families of triangles (Theorem 9, Corollary 10, Corollary 15, Theorem 17), shows that the exceptional set $T_f$ is totally disconnected (Theorem 5), and shows that minimal finite witnesses for the $(1,1,x)$-triangle grow without bound as $x\to1$ (Theorem 28).

Bears on

• Problem 173: Conjecture 2 is the problem's statement in the paper's terms, that a two-coloring misses at most one triangle (which must then be equilateral), and Conjecture 3 is equivalent to it by Theorem 1. The page poses the question and settles it for no coloring.