Hadwiger-Nelson problem (chromatic number of the plane)
Openhadwiger-nelson
Statement
Determine , the smallest number of colors required to color the points of the Euclidean plane such that no two points at distance exactly receive the same color. It is known that ; the task is to determine the exact value, or to strictly improve either the lower bound (currently ) or the upper bound (currently ).
Current frontier
Lower bound χ ≥ 5 via a 1581-vertex unit-distance graph (de Grey 2018, reduced to 509 vertices by Polymath 16 / Parts 2021); upper bound χ ≤ 7 via a hexagonal tiling (Isbell, c. 1950).
When this counts as solved
It's known that — closing this means pinning it down exactly with matching rigorous bounds. A single-direction strict improvement also counts as a breakthrough: an explicit finite unit-distance graph forcing (raising the lower bound), or an explicit 6-coloring of the plane with no two distance- points sharing a color (lowering the upper bound). Either alone is a top-journal result. Smaller witness graphs that still only force 5 colors, or measurable-coloring results that don't transfer, don't count.
Classification
value-determination