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Source. Theorem 8 with its proof, p. 570, and the table of four-point distance matrices that follows, 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 8 (p. 570, credited to Raphael M. Robinson). If five points can be found in the plane which determine only the distances $a,b,c,d$, where the distance $d$ (not necessarily distinct from $a,b,c$) occurs only once, and $a,b,c$ satisfy the triangle inequality, then $R(a,b,c)$ holds.

Proof pointer

P. 570. Since a proper two-coloring has bichromatic pairs at every distance (p. 570), place the five points with the two at distance $d$ oppositely colored; three of the five share a color, and that triple avoids the $d$-pair, so its sides lie in $\{a,b,c\}$. Theorem 1 and its corollaries then give $R_f(a,b,c)$.

To apply the theorem the paper classifies the four-point configurations with at most three distinct distances by their distance matrices and records which extend to a five-point set meeting the hypothesis (pp. 570--572); the extendable ones yield the families of Theorem 9. The paper states the extensions without proof.

Read depth. Claims checked: the statement and proof were read on p. 570; the classification on pp. 570--572 was read but not checked.

Bears on

• Problem 173: a sufficient condition for a single triangle to have a monochromatic congruent copy in every two-coloring of the plane, so that it is not the exceptional triangle of any coloring. It is the source of the families of Theorem 9.