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Source. Theorem 9 and the sentence after it, p. 572, with the configurations behind it, pp. 570--572, 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 9 (p. 572). $R(K)$ holds for all triangles $K=(a,b,c)$ which

• (i) have a $30^\circ$ angle;
• (ii) have a $150^\circ$ angle;
• (iii) are the sides and the circumradius of an isosceles triangle; the print adds "Satisfies $4a^2b^2-a^2c^2-b^4=0$";
• (iv) satisfy $c^2=a^2+2b^2$;
• (v) satisfy $a^2\pm ac-b^2=0$, $a\ne c$; the print adds that this includes $K$ with angles $(\alpha,2\alpha,180^\circ-3\alpha)$, $0<\alpha<60^\circ$, $\alpha\ne45^\circ$, and angles $(180^\circ-\alpha,180^\circ-2\alpha,3\alpha-180^\circ)$, $60^\circ<\alpha<90^\circ$, and $K=(a,2a,3a)$;
• (vi) satisfy $a^6-2a^4b^2+a^2b^4-3a^2b^2c^2+b^2c^4=0$;
• (vii) satisfy $a^4c^2+a^2b^4-5a^2b^2c^2+b^2c^4=0$.

Here $R(K)$ means that every two-coloring of $E^2$ has a monochromatic congruent copy of $K$ (Theorem 1). The letters in each relation are those of the corresponding four-point configuration on pp. 570--572; the print states the relations without saying which side is which; they are read here as holding for some labeling of the sides. In (iii) the relation holds when $b,b,c$ are the sides of the isosceles triangle and $a$ is its circumradius, matching the configuration on p. 571.

After the theorem the paper notes (p. 572) that the list includes the isosceles triangles with vertical angle $\theta=30^\circ$, $72^\circ$, $108^\circ$, $120^\circ$, $150^\circ$.

The introduction's partial list (p. 562) prints the relation of (vi) with a final term $b^2c^2$ where Theorem 9 and the configuration on p. 571 print $b^2c^4$.

Proof pointer

Pp. 570--572. Each item comes from a four-point configuration with at most three distances that the paper extends to five points meeting the hypothesis of Theorem 8: an arbitrary $30^\circ$ or $150^\circ$ triangle with its circumcenter, an isosceles triangle with its circumcenter, a parallelogram with a diagonal equal to a side, an isosceles trapezoid with one base equal to the legs, and two configurations of overlapping isosceles triangles.

Read depth. Claims checked: the statement was read clause by clause on p. 572 against the configurations on pp. 570--572; the extensions to five points were not checked.

Bears on

• Problem 173: every triangle of the seven families has a monochromatic congruent copy in every two-coloring of the plane, so none of them is the exceptional triangle of any coloring. The families are thin, each fixing an angle or a polynomial relation among the sides, and the theorem says nothing about whether a coloring can miss two other triangles.