Determine χ(ℝ^d): the chromatic number of the unit-distance graph in d-dimensional Euclidean space. Known bounds grow roughly exponentially in d, but the precise asymptotics are wide open even for the exponent base. In d = 2: 5 ≤ χ ≤ 7 (de Grey 2018 lower bound improvement).
Generalising Hadwiger–Nelson to higher dimensions immediately encounters Frankl–Wilson-style algebraic obstructions that give superpolynomial lower bounds, and exponential upper bounds from lattice colourings. The gap between upper and lower bounds grows exponentially with dimension — arguably worse relative precision than any other entry on our unsolved shelf.
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