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Statement

Setting (p. 4, Variant 4). The question is how many colors of the plane suffice to avoid a monochromatic congruent copy of a triangle with sides $1$, $a$ and $b$, for $0\le a,b\le1$ with $a+b>1$. For the equilateral triangle of side $1$, two colors suffice: alternating parallel half-open stripes of width $\sqrt3/2$, with the boundaries assigned correctly. The paper states a conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, cited from Graham's 2003 problem in The Open Problems Project, that three colors always suffice, and recalls that Aichholzer and Perz (2019) left some triangles with $a+b$ close to $1$ needing seven colors in their constructions. It also recalls (p. 4) the theorem of Currier, Moore and Yip (2024) that every two-coloring of the plane contains a monochromatic three-term arithmetic progression, a degenerate triangle with $a=b=0.5$.

Result (p. 4, Variant 4; Figure 4, p. 5; Section 4.4, p. 9; Contribution 3, p. 2). Placing the side of length $1$ as $AB$, Figure 4 shows the positions of the third vertex $C$ for which a coloring avoiding monochromatic copies of $ABC$ is known with three, four, five or six colors, before (Aichholzer and Perz) and after the paper's additions. The paper states that it formalized some of the stripe-based patterns its networks suggested and so obtained several new bounds that expand the regions where three to five colors suffice, which significantly reduces the part of the parameter space still requiring seven colors. It also states that Figure 7 of Aichholzer and Perz under-reported the four-color region, and that Figure 4 corrects this.

Status of the construction. The new bounds appear only as regions in Figure 4; the paper prints neither the colorings nor the inequalities on $a$ and $b$ that they cover, and says (p. 2) that a paper describing them is in preparation.

Numerical findings (Section 4.4 and Figure 8, p. 9; Appendix E, p. 19). Thousands of networks trained on sub-areas of the parameter region give a map of which triangles appear achievable with three to six colors, counting a point as achieved when the top $3\%$ of runs had under $0.1\%$ of sampled triangles monochromatic in the argmax coloring (Figure 16, p. 19, shows the maps for thresholds $1\%$, $0.1\%$ and $0.01\%$). The experiments needed more colors as one side shortened and the other approached length $1$, which the paper takes as evidence against the three-color conjecture. These maps are numerical, not theorems.

Source. Konrad Mundinger, Max Zimmer, Aldo Kiem, Christoph Spiegel and Sebastian Pokutta, Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?, Proceedings of the 42nd International Conference on Machine Learning, PMLR 267 (2025), arXiv:2501.18527, read in arXiv:2501.18527v3: Contribution 3 on p. 2, Variant 4 on p. 4, Figure 4 on p. 5, the loss for this variant on p. 7, Section 4.4 and Figure 8 on p. 9, Appendix E on p. 19. The edition read is identified on the source card.

Read depth. Claims checked: the setting, the stated outcomes and the figure captions were read on the print. No coloring was checked here, and the paper gives none to check. Nothing here is independently reviewed.

Proof pointer

None in this paper. The loss of Section 3.3 (p. 7) multiplies, for each color, the probabilities at $x$, at a point $y$ at distance $1$, and the average over the two points $z$ completing the triangle with $\lVert x-z\rVert=a$ and $\lVert y-z\rVert=b$.

Dependencies

None within the paper.

Bears on

• Problem 173: the problem asks whether every two-coloring of the plane contains a monochromatic congruent copy of every triangle but at most one. The only two-coloring here is the stripe coloring avoiding the unit equilateral triangle, the example the problem's commentary names, and the only two-color theorem is Currier, Moore and Yip's, recalled without proof; the new bounds are for three to five colors. The page proves no triangle unavoidable in two colors and bears on no instance of the problem.