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Source. Theorem 5, p. 565, with its proof, pp. 565--566, 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 5 (p. 565). For every two-coloring $f$ of $E^2$, the set $T_f$ is totally disconnected in $E^3$.

Here $T_f$ is the set of triples $(a,b,c)$ with $0\le a\le b\le c\le a+b$ such that $f$ has no monochromatic triangle with sides $a$, $b$, $c$ (pp. 563--564; see Theorem 1). The paper introduces the theorem as "something not as strong" as Conjecture 2, which says that $T_f$ has at most one element (p. 565).

Proof pointer

Pp. 565--566. If two triples $(a,b,c)$, $(a',b',c')$ with $a<a'$ lay in one component, Theorem 1 would put every equilateral triple $(d,d,d)$, $a\le d\le a'$, in $T_f$. Two like-colored points at the middle distance $a''=(a+a')/2$ then force a disc of radius $(a'-a)/2$ of the other color around the apex of their equilateral triangle. Rotating this along a supposed monochromatic circle of radius above $a''/2$ builds monochromatic annuli of unbounded thickness, which is impossible, so no such circle is monochromatic; two nearby oppositely colored pairs on a circle of radius $a''$ then give overlapping discs of opposite colors.

Read depth. Claims checked: the statement was read on the printed page; the proof was read for its structure only.

Bears on

• Problem 173: the problem asks that $T_f$ have at most one element for every $f$; the theorem shows only that $T_f$ contains no nontrivial connected piece, so no coloring misses a continuum of triangles along a curve. It leaves open whether $T_f$ can have two points.