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Statement

Conjecture 11.1.3 (p. 282). For every triangle $T$, $\mathbb{E}^2\not\xrightarrow{3}T$: there is a partition $\mathbb{E}^2=C_1\cup C_2\cup C_3$ in which no class contains a triangle congruent to $T$.

Here $\xrightarrow{r}$ is the relation of the Conjecture 11.1.1 page (p. 281), and the printed statement is the negated arrow with $3$ over it. Unlike Conjectures 11.1.1 and 11.1.2 it is a negative statement, and it covers every triangle, equilateral or not.

Scope

This is a conjecture, not a result proved in the chapter, and the chapter does not say whether degenerate triples count as triangles here. In three dimensions, Theorem 11.1.4(c) reports the opposite for three colors and every nondegenerate right triangle.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of J. E. Goodman, J. O'Rourke and C. D. Tóth (eds.), Handbook of Discrete and Computational Geometry, 3rd edition, CRC Press, Boca Raton, FL, 2017; the notation on p. 281 and the conjecture on p. 282. Pages are those printed on the edition named on the source card.

Read depth. Claims checked: the statement was read on the printed page, including the negation of the arrow.

Bears on

• Problem 173: context only. The conjecture concerns three-colorings, and Problem 173 two-colorings; the chapter draws no connection between the two, and the conjecture decides nothing about the problem.