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  "body": "\n## Problem\n\nLet $G$ be the [[graph|00C0BR7D7H]] whose vertices are the points of the plane $\\R^2$.\nTwo points $p$ and $q$ are adjacent exactly when their [[Euclidean distance|0NPPGHQNXQ]] is $\\norm{p - q} = 1$.\nDetermine the [[chromatic number|0VHJRXNAAS]] $\\chi(G)$ of this graph.\nEquivalently, colour every point of the plane so that any two points at distance $1$ get different colours.\nFind the least number of colours that makes this possible.\nThis number is called the chromatic number of the plane.\n\n## Remarks\n\nThe problem is open.\nThe known bounds are $5 \\le \\chi(G) \\le 7$.\nThe upper bound $7$ comes from a tiling of the plane by regular hexagons of diameter slightly less than $1$, coloured with $7$ colours in a periodic pattern.\nThe lower bound $4$ follows from a finite unit-distance graph with $7$ vertices, the Moser spindle, which has no proper [[colouring|0NCPXH194Y]] with $3$ colours.\nIn 2018 Aubrey de Grey found a finite unit-distance graph that needs $5$ colours, which raised the lower bound to $5$.\nBy the de Bruijn–Erdős theorem (assuming the axiom of choice), $\\chi(G)$ equals the largest chromatic number of a finite subgraph of $G$.\n\n## Explanations\n\n### L1 human\n\nPaint every point of the plane with some colour.\nThe rule is that two points exactly one unit apart must get different colours.\nWhat is the smallest number of colours that makes this possible?\nThree colours are not enough, and seven colours are enough.\nToday we know that at least five are needed, but whether the answer is $5$, $6$ or $7$ is unknown.\n",
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      "posed_by": "Edward Nelson"
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  "msc": [
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  "name": "Hadwiger–Nelson problem",
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