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Source. Conjecture 2 on p. 1 and the coloring of Figure 1 on p. 2 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

Conjecture 2 (p. 1, quoted). "For any triangle $T$, there exists a 3-coloring of $\mathbb{E}^2$ without a monochromatic copy of $T$."

A copy means a congruent copy, as in the paper's definitions on p. 1. The paper does not prove the conjecture.

The (a,a,2a) coloring (p. 2). The authors report a three-coloring of the plane with no monochromatic congruent copy of the degenerate triangle with sides $a,a,2a$, that is, of three equally spaced collinear points at spacing $a$. The plane is tiled by hexagons of diameter $2a$, each half-open, and the hexagons receive three colors in the pattern of the paper's Figure 1. The paper gives the coloring by its figure and does not write out the verification.

The paper adds (p. 2) that the three-coloring by alternating half-open strips avoids a large class of triangles but not all, and that for any collinear set 16 colors suffice to avoid a monochromatic copy in $\mathbb{E}^n$ for every $n$ (Straus, its reference [27]), whether 16 is best possible being open.

Read depth. Claims checked: the conjecture and the description of the coloring were read on pp. 1--2 of the preprint; the coloring's correctness was not checked here.

Bears on

No Erdős problem page is linked. The conjecture concerns three-colorings; the corpus's triangle question, Problem 173, concerns two-colorings, on which Conjecture 2 makes no claim.