---

Statement

Definition (p. 5). Given a colouring $F$ of the plane, a triangle $T=xyz$ is a monochromatic limit triangle when some monochromatic set $\{x_1,y_1,z_1,x_2,y_2,z_2,\ldots\}$ has $x_n\to x$, $y_n\to y$, $z_n\to z$ with every triangle $T_n=x_ny_nz_n$ similar to $T$.

Theorem 1 (p. 5), credited to Nielsen, the paper's [10]: "Let $F$ be a two-colouring of the plane and let $T$ be a triangle. Then $F$ admits a monochromatic limit triangle congruent to $T$."

The theorem is quoted, not proved, in this paper. The paper's [10] is M. J. Nielsen, Approximating monochromatic triangles in a two-colored plane, Acta Math. Hungar. 74 (1997), no. 4, 279-286. The library holds no card for that paper, and the statement here is the form this paper prints, not checked against Nielsen's.

Source. J. Grytczuk, K. Junosza-Szaniawski, J. Sokół, K. Węsek, Fractional and $j$-fold coloring of the plane, Discrete Comput. Geom. 55 (2016), 594-609, doi:10.1007/s00454-016-9769-3; read in arXiv:1506.01887v2 (5 October 2015), Theorem 1 and the definition before it on p. 5 of that version. The source card records the edition.

Read depth. Claims checked: the definition and the quoted statement were read clause by clause on the print.

Proof pointer

None in this paper; see Nielsen's paper cited above.

Dependencies

None in this paper. The proof of Theorem 2 uses it.

Bears on

• Problem 173: the problem asks whether every two-colouring of the plane contains a monochromatic congruent copy of every triangle with at most one exception. Theorem 1 gives, for every triangle, monochromatic triangles similar to it that converge to a triangle congruent to it; it does not give a monochromatic congruent copy of any triangle, and this paper proves nothing further on the problem.