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Source. Conjecture 1 on p. 1 of Ron Graham and Eric Tressler, Open problems in Euclidean Ramsey theory, in A. Soifer (ed.), Ramsey Theory: Yesterday, Today, and Tomorrow, Progress in Mathematics, Birkhäuser (2011), 115--120, doi:10.1007/978-0-8176-8092-3_7. Page numbers here are those of the authors' preprint, the edition read, as identified on the source card.

Statement

Setting (p. 1). For a finite set $X$, $\mathrm{Cong}(X)$ is the set of sets congruent to $X$ under a Euclidean motion. For a set $S$, $S\xrightarrow{r}X$ means that every $r$-coloring of $S$ contains a monochromatic member of $\mathrm{Cong}(X)$; a triangle is identified with its set of three vertices.

Conjecture 1 (p. 1; the paper cites the third Euclidean Ramsey paper of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, its reference [9]). For every non-equilateral triangle $T$, $\mathbb{E}^2\xrightarrow{2}T$.

The paper does not prove or refute it. Context it reports (p. 2): for each equilateral triangle $T$ the coloring of the plane by alternating half-open strips of height the altitude of $T$ avoids a monochromatic copy of $T$, so the hypothesis "non-equilateral" cannot be dropped; Jelínek, Kynčl, Stolař and Valla showed there are infinitely many two-colorings avoiding a given equilateral triangle, and that Conjecture 1 holds for colorings in which one color class is open and the other closed; the conjecture is known for many classes of triangles (the paper's reference [9]), and for right triangles by Shader (reference [22]). None of these is proved in the survey.

Read depth. Claims checked: the statement and the reported context were read clause by clause on pp. 1--2 of the preprint.

Bears on

• Problem 173: the problem asks that every two-coloring of the plane contain a monochromatic congruent copy of all but at most one triangle. Conjecture 1 would confine the exceptions in any two-coloring to equilateral triangles; it does not say how many equilateral triangles one coloring can avoid, so it does not by itself give the problem's "at most one". The strip coloring the paper describes shows that an exception can occur. The paper proves neither.