Chromatic number of 3-space: improve the bounds on $\chi(\mathbb{R}^3)$
Statement
Color every point of $\mathbb{R}^3$ so that no two points at Euclidean distance exactly $1$ receive the same color. The chromatic number $\chi(\mathbb{R}^3)$ is the least number of colors admitting such a coloring. Improve the known bounds on $\chi(\mathbb{R}^3)$: raise the lower bound above $6$, or lower the upper bound below $15$, with a verifiable certificate.
Acceptance. ADVANCES: a finite unit-distance graph $G\subset\mathbb{R}^3$ (explicit vertex coordinates) with a machine-checkable proof (e.g. DRAT/UNSAT certificate) that $G$ is not $6$-colorable, raising the lower bound to $\ge 7$; or an explicit coloring rule for $\mathbb{R}^3$ with a proof it uses $<15$ colors and no color class realizes distance $1$, lowering the upper bound. FULLY RESOLVES: matching bounds pinning $\chi(\mathbb{R}^3)$. Provide coordinates/coloring plus a verification script; lower-bound certificates are finite and checkable.
Background
The current bounds are $6 \le \chi(\mathbb{R}^3) \le 15$. Lower bound $6$: Nechushtan ('On the space chromatic number', 2002), via a finite unit-distance graph in $\mathbb{R}^3$ that is not $5$-colorable; de Grey (2020) exhibited a $59$-vertex $6$-chromatic unit-distance graph in $\mathbb{R}^3$. Upper bound $15$: Coulson, and independently Radoičić–Tóth, via explicit periodic tiling/colorings. This is the space analogue of the Hadwiger–Nelson problem for the plane, where de Grey (2018) pushed the plane lower bound $4\to5$ using a $\sim1500$-vertex SAT-verified unit-distance graph (the plane value remains in $\{5,6,7\}$). The $3$-space analogue has seen far less computational assault; the gap $[6,15]$ is wide. An attacker must bring: (lower bound) a generator of large explicit unit-distance graphs in $\mathbb{R}^3$ (Minkowski sums / rotations of small rigid gadgets) plus a SAT/CP solver to certify non-$k$-colorability — any such graph is a finite, machine-checkable certificate; (upper bound) an explicit periodic coloring of $\mathbb{R}^3$ with a proof it uses $<15$ colors and no color class contains a unit-distance pair. Distinguish $\chi(\mathbb{R}^3)$ (arbitrary colorings, treated here) from the measurable/tiling chromatic number and from sphere-coloring variants, which have separate, recently-moving bounds.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Nechushtan, On the space chromatic number (lower bound 6) | paper |
| REF-02 | On the chromatic numbers of 3-dimensional slices | arxiv |
| REF-03 | de Grey, The chromatic number of the plane is at least 5 (SAT method precedent) | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.