The signed cube needs linearly many colors

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This open claim is recorded by Plasma AI. By L263 it implies the exponential bound $f(k)\le2^{k/c+1}$ that Problem 483 asks for, and by L265 the uniform Bohr-cell hypothesis would imply it. That hypothesis is refuted in L266, independently of this claim, as the research sketch explains. Whether the exponential Schur bound implies the claim is unresolved here.

Equivalently, $\chi_r$ is the least number of colors in a triangle-free coloring of the edges of the complete graph on $\{0,1\}^r$ in which the color of $\{x,y\}$ depends only on $\pm(y-x)$. General triangle-free colorings of that graph need only $o(r)$ colors, by the accepted claim on Problem 183, so the claim asserts that invariance under differences costs a factor that grows without bound.

Standing

Open in these notes; neither a proof nor a counterexample is established. The signed-cube page records the evidence: $\chi_1,\ldots,\chi_7=1,2,3,3,4,5,5$; the bound $\chi_r\le0.584\,r+O(1)$ through binary values and the Schur colorings of Ageron et al.; the bound through sum-free colorings of $\mathbb F_2^r\setminus\{0\}$, whose linear growth is itself an open conjecture that Hunter and Pohoata record (arXiv:2309.02424v2, p. 11); subadditivity; and the Ramsey lower bound $\chi_r\ge(1-o(1))\,r\log2/\log r$, which no argument through general triangle-free colorings can raise above $o(r)$. So any constant $c$ in the claim is at most about $0.584$. Two web searches on 2026-10-15 found no treatment of this coloring number; they are not exhaustive.

Remaining obligations

• Prove or refute the claim. The refutation of uniform Bohr-cell avoidance does not decide it: that construction colors a slice of the cube, not the whole cube.
• A family of sum-free colorings of $\mathbb F_2^n\setminus\{0\}$ by $k$ colors with $n/k\to\infty$ would refute the claim through the support map; the claim needs the conjectured linear bound for those colorings.
• The first value not determined by the arguments here is $\chi_8\in\{5,6\}$; a five-coloring would also give $R_5(3)>256$.