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Statement

Setting (p. 3, Variant 2). A $k$-coloring of the plane has coloring type $(d_1,\ldots,d_k)$ if color $i$ does not realize distance $d_i$. Soifer's continuum of six-colorings is the set of $d$ for which a six-coloring of type $(1,1,1,1,1,d)$ exists; before this work it was known to contain every $d$ with $\sqrt2-1\le d\le1/\sqrt5$ (Hoffman and Soifer 1996, Soifer 1994).

Result (p. 4, Variant 2; Contribution 1, p. 2). Two six-colorings:

• The first is parameterized by $d$ and has type $(1,1,1,1,1,d)$ for $0.354\le d\le0.553$ (Figure 3, p. 4, drawn for $d=0.45$).
• The second is a single coloring in which the sixth color avoids every distance $d$ with $0.418\le d\le0.657$ and the other five avoid distance $1$ (Figure 1, p. 1, and Figure 11, p. 15).

Together they give six-colorings of type $(1,1,1,1,1,d)$ for every $d$ in $[0.354,0.657]$; the paper compares this with the earlier range, which it writes as $[0.415,0.447]$ on p. 2.

Status of the construction. The colorings are shown in the figures, and the paper states (pp. 2 and 4) that their complete description is given in Mundinger, Pokutta, Spiegel and Zimmer, Extending the continuum of six-colorings, Geombinatorics Quarterly (2024). This paper does not prove the ranges.

Numerical findings (Section 4.2, p. 8; Appendix C, pp. 15-17). Networks for types $(1,1,1,1,d_1,d_2)$ found low-conflict regions near $d_1$ or $d_2$ about $0.5$ with the other equal to $1$, and near $(0.5,0.5)$. A search for five-colorings of type $(d_1,\ldots,d_5)$ reached a conflict rate of about $5\%$ (4.9% on p. 8) at $d_1=1$, $d_2\approx d_3\approx1$, $d_4\approx d_5\approx0.56$; the paper reads its failure to find a conflict-free five-coloring as evidence that the polychromatic number of the plane may be six. These are numerical observations, 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: Figure 1 on p. 1, Contribution 1 on p. 2, Variant 2 and Figure 3 on pp. 3-4, Section 4.2 on p. 8, Appendix C on pp. 15-17. The edition read is identified on the source card.

Read depth. Claims checked: the stated ranges and attributions were read on the print. The colorings were not checked here, and their proof lies in the separate 2024 paper. Nothing here is independently reviewed.

Proof pointer

None in this paper; see the 2024 paper named above. The networks of Section 3.3 (pp. 6-7) take $d$ as an extra input, which let the authors follow colorings continuously in $d$ (Figure 12, p. 15) before formalizing them.

Dependencies

None within the paper.

Bears on

• Problem 508: a coloring of type $(1,1,1,1,1,d)$ with $d\ne1$ is not a proper six-coloring for unit distance, so the result gives no bound on the chromatic number of the plane.