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Source. Theorem 1, the remark after it and Corollaries 2 and 3, p. 563; the sets $T$ and $T_f$, pp. 563--564; Corollary 4, p. 564; 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.

Notation

The paper works in the plane $E^2$ with two colors and three-point sets $K$ (p. 559). $R(K)$ says that every two-coloring of $E^2$ has a monochromatic $K'$ congruent to $K$; for a fixed two-coloring $f$, $R_f(K)$ says that some $K'$ congruent to $K$ is monochromatic under $f$ (p. 562). $R(a,b,c)$ and $R_f(a,b,c)$ refer to the triangle with sides $a,b,c$. $R_f(\bar a,b,c)$ says that $f$ has an $(a,b,c)$-triangle whose $a$-side has both endpoints of one color and whose third point has the other color (p. 561). A two-coloring is proper when it is not a one-coloring (p. 573).

The paper puts $T=\{(a,b,c):0\le a\le b\le c\le a+b\}$ and, for each two-coloring $f$, lets $T_f$ be the set of triples $(a,b,c)\in T$ with no monochromatic triangle of sides $a$, $b$, $c$ (pp. 563--564).

Statement

Theorem 1 (p. 563). Let $K$ be a triangle with sides $a$, $b$, $c$, and let $K_a$, $K_b$, $K_c$ be the equilateral triangles of sides $a$, $b$, $c$. Then $R_f(K)$ holds if and only if at least one of $R_f(K_a)$, $R_f(K_b)$, $R_f(K_c)$ holds.

The paper calls it a strengthening of Theorem 8 of Part I (Euclidean Ramsey Theorems I). In the notation above it says that $(a,b,c)\in T_f$ exactly when $(a,a,a)$, $(b,b,b)$ and $(c,c,c)$ all lie in $T_f$.

Remark (p. 563, credited to R. M. Robinson). The six copies of $K$ in the proof are like-oriented, so the proof gives more: if $K$ has a monochromatic like-oriented congruent copy under $f$, it also has a monochromatic opposite-oriented one. The paper does not know the analogue for bichromatic copies.

Corollaries (pp. 563--564).

• Corollary 2: if $K$ has sides $a,a,b$ and $R_f(K)$ holds, then $R_f(K^*)$ holds for every triple $K^*$ with sides $a,b,c$ where $|a-b|\le c\le a+b$.
• Corollary 3: if $R_f(K_a)$ fails but $R_f(K)$ holds for a triple $K$ with sides $a,b,c$, then $R_f(K^*)$ holds for every triple $K^*$ with sides $b,c,d$, $|b-c|\le d\le b+c$.
• Corollary 4: let $K$ be an $(a,a,b)$-triangle with $R(K)$, let $f$ be a two-coloring of $E^2$ and suppose $(c,d,e)\in T_f$; then $(bc/a,bc/a,bc/a)\notin T_f$ and $(ac/b,ac/b,ac/b)\notin T_f$. The paper notes after the proof that $d$ or $e$ can replace $c$.

The paper also observes (p. 564) that by Theorem 1, Conjecture 3 is equivalent to $T_f\subset\{(a,a,a)\mid a>0\}$ for every $f$.

Proof pointer

P. 563, from Figure 1 (p. 564): eight points $A,\ldots,H$ carry six triangles with sides $a,b,c$ and six equilateral triangles, two of each side $a$, $b$, $c$, arranged so that, as in Theorem 8 of Part I, a monochromatic one among the equilateral six forces a monochromatic one among the other six; the converse is the symmetric argument.

Read depth. Claims checked: the statements were read clause by clause on the printed pages; the proof was read for its structure only.

Bears on

• Problem 173: the theorem reduces the problem to equilateral triangles. A two-coloring misses a triangle exactly when it misses the equilateral triangles on all three of its side lengths, so the problem is equivalent to the statement that no two-coloring of the plane misses equilateral triangles of two different sides. The theorem decides no triangle by itself.