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Claim. Aubrey D. N. J. de Grey, The chromatic number of the plane is at least 5, Geombinatorics 28 (2018), no. 1, 18–31; posted as arXiv:1804.02385 on 8 April 2018; carded at grey_2018_chromatic_number_plane_is_at_least. The paper presents a family of finite unit-distance graphs in the plane that admit no proper $4$-coloring; the smallest it reports has $1581$ vertices. Since a proper coloring of the plane restricts to a proper coloring of every unit-distance graph drawn in it, the chromatic number asked for by Problem 508 satisfies $\chi(\mathbb R^2)\ge5$, the first improvement of either Hadwiger–Nelson bound since the bounds $4\le\chi\le7$ of 1950. The construction starts from the seven-vertex hexagonal unit-distance graph $H$, whose $4$-colorings fall into four types, two with a monochromatic triple of vertices and two without; a graph $L$ of $52$ copies of $H$ forces some copy to carry a monochromatic triple in every $4$-coloring, and a graph $M$, checked by a custom backtracking search, admits no $4$-coloring in which its central copy of $H$ carries one. Assembling copies of $M$ along $L$ gives a non-$4$-colorable graph on $20425$ vertices, which deleting and adding vertices reduces to $1581$ vertices; the paper reports that others confirmed with SAT solvers that this graph has no $4$-coloring, as recorded on the Section 5.1 page.

Covers. The exclusion of four colors, $\chi(\mathbb R^2)\ge5$, and no more: the paper gives no upper bound and does not determine $\chi$. The bound is superseded by the accepted exclusion of five colors on OpenAI's claim page, which gives $6\le\chi(\mathbb R^2)\le7$.

Depends on. No page of this wiki.

Acceptance. Refereed: Geombinatorics published the paper. Later papers found smaller non-$4$-colorable unit-distance graphs and other proofs of the bound, recorded on the problem page. The site's curator credits the lower bound five to this paper in the problem's commentary, but the site labels the problem OPEN, so that credit is context and not reviewed evidence.