How many colors do you need to paint all of eight-dimensional Euclidean space so that no two points exactly one unit apart share a color? The answer was known to be at least 17. Jaan Parts pushes it to 25.
This is the chromatic number problem for R^n, and it is deceptively simple to state. In two dimensions, the answer is between 5 and 7 — a gap that has persisted since the problem was posed in 1950. Every additional dimension makes the problem harder, because the constraint set (all unit-distance pairs) grows explosively while the tools for proving lower bounds (finite point configurations with forced colorings) must be constructed one at a time.
The method is computational: find a finite set of points in R^8 where 24 colors provably fail. The points must be chosen so that the unit-distance graph among them has chromatic number at least 25. This is a search problem in a space that is both enormous (all possible finite subsets of R^8) and fragile (adding one point can change everything).
What makes dimension 8 special is the E8 lattice, the densest sphere packing in eight dimensions, whose kissing number (240) and exceptional symmetry provide a scaffolding for the point configurations that force high chromatic numbers. The lattice is doing structural work — not just sitting there as a curiosity of high-dimensional geometry, but actively constraining what's possible.
The gap between the lower bound (now 25) and the upper bound (presumably much higher) remains wide. The improvement is incremental. But incremental improvements in lower bounds are the hard direction — upper bounds give you a coloring, while lower bounds require proving that no coloring works. The asymmetry is fundamental: construction is easier than obstruction.