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Source. Conjecture 1 and the strip coloring before it, pp. 559--560, and the remark after it, p. 560, 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 1 (p. 560), as posed: "The only 2-colorings of $E^2$ for which there are no monochromatic equilateral triangles of side $d$ are colorings in alternate strips of width $(\sqrt3/2)d$, as above, except for some freedom in coloring the boundaries between the strips."

The coloring "above" (pp. 559--560) colors red the union over all integers $n$ of the half-open strips $nd\sqrt3\le y<(n+\tfrac12)d\sqrt3$ and blue the rest; it has no monochromatic equilateral triangle of side $d$, and the paper notes that some changes on the boundary lines keep that property, for instance recoloring each of the points $n(\tfrac12d,\tfrac{\sqrt3}{2}d)$.

The paper states without proof (p. 560) that in such a strip coloring the equilateral triangle of side $d$ is the only one that fails to occur monochromatically, so a strip coloring avoids only one size of equilateral triangle. It calls Conjecture 2 a weaker conjecture that may hold even if Conjecture 1 fails.

Status in the paper

Posed, not proved. The paper proves nothing toward it beyond the example.

Bears on

• Problem 173: with the paper's remark that a strip coloring misses only the equilateral triangle of side $d$ and with Theorem 1, Conjecture 1 would imply that every two-coloring misses at most one triangle, which is the problem's statement. The conjecture is a stronger structural statement and is not proved here.