Erdős et al.'s conjecture on avoiding triangles in two-colored planes

Let $T$ be a triangle in the Euclidean plane, and let a coloring be a partition of $\mathbb{R}^2$ into two color classes. A coloring contains $T$ if it has a monochromatic copy of $T$ obtained by translations and rotations; otherwise, it avoids $T$. Let $\chi^*$ be the coloring by alternating half-open horizontal strips of width $\frac{\sqrt{3}}{2}$, with a point $(x,y)$ black if and only if $n\sqrt{3}<y\leq\left(n+\frac{1}{2}\right)\sqrt{3}$ for some integer $n$. Erdős et al.'s conjecture. For every triangle $T$ and every coloring $\chi$, if $\chi$ avoids $T$, then $T$ is an equilateral $(l,l,l)$-triangle and $\chi$ is equal to an $l$-times scaled copy of $\chi^*$, up to possible modifications of the colors of points on the boundary of the strips. The paper presents a counterexample to this conjecture and gives a broader class of colorings avoiding the unit triangle, so this conjecture is refuted.