Signed cubes as a sufficient condition for exponential Schur growth

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The Bohr-cell recursion refines $[0,N]$ through cells of growing rank. After many steps a Bohr cell can be a digit cube, and on a digit cube the Schur structure of the differences is exactly the vector addition of a signed cube. This page collects what is known about the coloring number of that cube and how it bears on the route and on the target. The results are author-recorded by Plasma AI; no independent examination is recorded. Two web searches on 2026-10-15 for Schur-type colorings of $\{-1,0,1\}^n$ under vector addition found no treatment of $\chi_r$; that search is not exhaustive.

The signed cube

Let $Q_r=\{-1,0,1\}^r\setminus\{0\}$, and let $Q_r^+$ be the set of $v\in Q_r$ whose last nonzero coordinate is $1$, so that $Q_r=Q_r^+\cup(-Q_r^+)$. A triple of $Q_r^+$ is $(v,w,u)$ with $v,w,u\in Q_r^+$ and $v+w=u$ coordinatewise; $v=w$ never occurs, since $2v\notin Q_r$. Let $\chi_r$ be the least number of colors in a coloring of $Q_r^+$ with no monochromatic triple.

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 $\pm(y-x)$. Given a coloring of $Q_r^+$, color $\{x,y\}$ by the positive representative of $y-x$; a triangle $x,y,z$ gives $(y-x)+(z-y)=z-x$, and moving negative terms across turns it into a triple of positive representatives, since a sum of two vectors of $Q_r^+$ that lies in $Q_r$ lies in $Q_r^+$. Conversely every triple $(v,w,u)$ is a triangle: no coordinate has $(v_c,w_c)=\pm(1,1)$, so coordinatewise one can choose $x_c,y_c,z_c\in\{0,1\}$ with $y-x=w$ and $z-y=v$.

For $b\ge4$ the base-$b$ value $\iota_b(v)=\sum_iv_ib^{i-1}$ is injective, positive exactly on $Q_r^+$, and carries triples exactly to the solutions of $x+y=z$ among its values; for $b=q+1$ with $q\ge3$, the $0/1$ digit cube in base $b$ is the Bohr cell $\{y:\{y/b^l\}\in[0,1/q)$ for $l\le r\}$, and its positive differences are $\iota_b(Q_r^+)$. These facts are Lemma 2 of the proof of L265.

Two implications

• A linear lower bound for $\chi_r$ would settle the target: L263 shows that $\chi_r\ge cr$ for every $r$ gives $f(k)\le2^{k/c+1}$, by pulling Schur colorings of $[1,2^r-1]$ back along the binary value. That linear bound is the open claim L264.
• The Bohr-cell hypothesis $(B_\rho)$ would force it: if every Schur-free coloring satisfies $(B_\rho)$ with $\rho\le1/3$, then $\chi_r\ge r\log2/\log\lceil1/\rho\rceil$, by part (2) of L265, which plants a cube coloring on the digit cube; the threshold form of the hypothesis would force a linear bound as well. The universal hypothesis is refuted in L266; its failure does not refute the signed-cube bound.

Whether the exponential bound implies $\chi_r\ge cr$ is unresolved here. A Schur coloring of an interval pulls back to the cube; this gives no extension theorem for cube colorings.

What is known about the coloring number

Upper bounds.

1. $\chi_r\le H(2^r-1)$, where $H(n)$ is the least number of colors in a coloring of $[1,n]$ with no monochromatic $x+y=z$ ($x=y$ allowed); this is the pullback in the proof of L263. Schur's tripling coloring gives $\chi_r\le\lceil(r+1)\log2/\log3\rceil$, and inequality (6) of Ageron et al. gives $\chi_r\le(5\log2/\log380)\,r+O(1)\approx0.584\,r$.
2. If $\mathbb F_2^r\setminus\{0\}$ is a union of $k$ sum-free sets, then $\chi_r\le k$. The support map $\sigma(v)=\{i:v_i\ne0\}\in\mathbb F_2^r$ is nonzero on $Q_r$ and satisfies $\sigma(v)+\sigma(w)=\sigma(u)$ on every triple, since a coordinate where both $v$ and $w$ are nonzero has opposite signs there and vanishes in $u$.
3. $\chi_{a+b}\le\chi_a+\chi_b$ and $\chi_r\le\chi_{r+1}$. For the first, color $(v',v'')\in Q_{a+b}^+$ by a coloring of $Q_b^+$ at $v''$ when $v''\ne0$, and by a coloring of $Q_a^+$ with a disjoint palette at $v'$ when $v''=0$; a triple with both summands in the first palette and $u''\ne0$ is a triple of $Q_b^+$, and one with $u''=0$ changes palette. For the second, append a zero coordinate.
4. The coloring by the position of the last nonzero coordinate has no monochromatic triple, since two such vectors sum to a $2$ there, so $\chi_r\le r$.

Ramsey lower bound. The equivalent description gives a triangle-free $\chi_r$-coloring of $K_{2^r}$, so $2^r<R_{\chi_r}(3)$. With $R_k(3)\le ek!+1$, the classical bound that Eliahou's card records, this gives $\chi_r\ge(1-o(1))\,r\log2/\log r$. By the accepted claim on Problem 183, $R_k(3)^{1/k}\to\infty$, so no argument through general triangle-free colorings can give more than $o(r)$: they need only $o(r)$ colors on $K_{2^r}$, and L264 asks whether invariance under differences forces $\Omega(r)$.

Sum-free colorings of $\mathbb F_2^n$. Let $n_2(k)$ be the largest $n$ such that $\mathbb F_2^n\setminus\{0\}$ is a union of $k$ sum-free sets. By upper bound 2, $\chi_{n_2(k)}\le k$, so L264 forces $n_2(k)=O(k)$, and so does $(B_\rho)$ by L265: $n_2(k)\le k\log\lceil1/\rho\rceil/\log2$. The elementary bounds used here are $n_2(3)=4$, from the three cubic-residue classes of $\mathrm{GF}(16)$ and $2^5\ge R_3(3)$; $n_2(k)\ge4\lfloor k/3\rfloor$ by superadditivity; and the Cayley coloring of $K_{2^n}$ gives $2^{n_2(k)}<R_k(3)\le ek!+1$, so $n_2(k)=O(k\log k)$. A coloring of $\mathbb F_2^n\setminus\{0\}$ into sum-free sets is exactly a coloring of the points of $\mathrm{PG}(n-1,2)$ with no monochromatic line. A. Bishnoi, W. Cames van Batenburg and A. Ravi, The chromatic number of finite projective spaces, arXiv:2512.01760v3 (24 May 2026; abstract only, not held by the library), state that the least number of colors $\chi_2(n)$ for such a coloring satisfies $\chi_2(n)\le\lfloor2n/3\rfloor+1$ for $n\ge2$, with equality for $n\le7$, and $\chi_2(n)\ge(1-o(1))\,n/\log n$. Hence $n_2(k)\ge\lceil3k/2\rceil-1$, which improves the superadditive bound; the abstract leaves a linear lower bound for $\chi_2(n)$ open. Z. Hunter and C. Pohoata, On off-diagonal Ramsey numbers for vector spaces over $\mathbb F_2$, arXiv:2309.02424v2 (23 December 2024), Section 4 and its footnote 2 on p. 11 (read in the arXiv PDF text; not held by the library), discuss the same covering question for the punctured spaces $\mathbb F_p^n\setminus\{0\}$: $O(n\log p)$ sum-free sets suffice, the Ramsey connection rules out $o(n/\log n)$, and $\Omega(n\log p)$ sets are conjectured to be needed, a conjecture they tie to the growth of the multicolor triangle Ramsey numbers. An exponential bound $R_k(3)\le C^k$ would have given $n_2(k)\le k\log_2C$; the accepted claim on Problem 183 rules that bound out, so that argument for the linear conjecture is unavailable. No linear bound or superlinear lower family for $n_2$ is established here. Their Theorem 4.1 finds a subspace avoiding all of $r$ solution-free sets once $n>C_pr(\log r+d)^7$, a subspace analogue of the Bohr-cell recursion with a polylogarithmic loss per color.

For the superadditivity used above, color a nonzero vector in a direct sum by its first block when that block is nonzero, and otherwise by its second block, using disjoint palettes. A monochromatic sum staying in one palette would contradict the corresponding block coloring; a sum whose first block cancels changes palette. Thus color counts add while dimensions add.

Small values. $\chi_1,\ldots,\chi_7=1,2,3,3,4,5,5$.

• Lower bounds: $\chi_2\ge2$, since $(1,0)+(0,1)=(1,1)$ is a triple; $\chi_3\ge3$ by exhaustion (also $8\ge R_2(3)=6$); $\chi_5\ge4$ from $32\ge R_3(3)=17$; $\chi_6\ge5$ from $64>62\ge R_4(3)$ (Fettes, Kramer and Radziszowski); the others by monotonicity. The values $R_2(3)=6$ and $R_3(3)=17$ are the classical ones recorded on Eliahou's card.
• Upper bounds: $\chi_3\le3$ by the last-nonzero-coordinate coloring; $\chi_4\le3$ from the cubic-residue classes of $\mathrm{GF}(16)$, which are sum-free in $\mathbb F_2^4$, through the support map; $\chi_5\le4$ and $\chi_6\le5$ from the tripling colorings of $[1,40]$ and $[1,121]$ through the binary value; $\chi_7\le5$ from a Schur-free five-coloring of $[1,133]$, the composition (Lemma A of the refutation of L260) of one color on $[1,1]$ with a four-coloring of $[1,44]$, through the binary value, since $2^7-1\le133$; Exoo's five-coloring of $[1,160]$ would serve as well.
• $\chi_8\in\{5,6\}$ is the first value these arguments leave undecided: the tripling coloring of $[1,364]$ gives 6, while $S(5)=160<255$ and the bound $R_5(3)\le307$ that Eliahou's card records decide nothing. A five-coloring would give $R_5(3)>256$ by the Ramsey lower bound above, far beyond the bound $R_5(3)\ge162$ recorded there; in general a cube coloring that beats the binary-value pullback improves the lower bound $R_k(3)\ge S(k)+2$.

The retained evidence checks the base-$b$ facts for $r\le5$, the digit-cube cells for $r\le4$, the binary-value and support maps for $r\le6$, the exhaustion for $\chi_3$, and the colorings behind $\chi_4\le3$, $\chi_5\le4$, $\chi_6\le5$ and $\chi_7\le5$, with the four-coloring of $[1,44]$ retained as an input.

The hierarchy and its scope

For $N=2^r$, with $R^{-1}(N)$ the least $k$ with $R_k(3)>N$,

$$R^{-1}(2^r)\le\chi_r\le\min\bigl(H(2^r-1),\ \min\{k:n_2(k)\ge r\}\bigr).$$

The left end is $o(r)$. The exponential conjecture asserts $H(2^r-1)\ge cr$. A linear bound for $\chi_r$ would imply both that conjecture and $n_2(k)=O(k)$. The uniform Bohr-cell hypothesis would imply the linear bound, but its refutation is logically independent of whether the linear bound is true.

The refutation in L266 uses a digit-sum-one slice of a digit cube, whose differences encode arbitrary triangle-free edge colorings. Coloring that slice does not supply a coloring of all of $Q_r^+$. Hence it neither constructs sublinear $\chi_r$ nor resolves the Schur target. Conversely a sublinear family for $\chi_r$ would require a further extension theorem to give interval colorings. A proof of linearity would prove the Schur upper bound by L263.

The missing Cayley transfer

A coloring of the nonzero elements of a finite abelian group $G$ into sum-free classes gives a triangle-free edge coloring by differences, provided opposite elements have the same color. For $G=\mathbb F_2^r$ this symmetry is automatic, and the support map above would transfer the coloring to $Q_r^+$. Thus an accepted Ramsey lower construction on $\mathbb F_2^r$ with a difference-only coloring and $k=o(r)$ would refute L264. The accepted theorem for arbitrary edge colorings does not assert this additional structure.

The held reconstruction of OpenAI's August 6, 2026 report, Chapter 9, Proposition 3.1, printed pp. 233–234, describes the actual coloring rule. Its vertices are recursive blocks $V_P$ indexed by a separated family of palettes $P$. Internal copies use colors outside $P$. For each ordered block pair $P<Q$, the construction chooses lists in $Q\setminus P$ and $P\setminus Q$. A cross edge's color is selected by equalities involving separate proper-color labels of its two endpoints and the fixed maps $f,g$ of Lemma 2.2. The palette lemma supplies a separated family of subsets, not a group operation on blocks. The final theorem uses the product of the block counts to obtain its size bound.

These pages provide no identification of vertices with an abelian group under which the edge color depends only on the difference. In particular, the choices depend on the block pair and on the endpoint labels; no translation-equivariance identity is proved. Even regularity of each individual color graph, a necessary condition for a Cayley coloring, is not part of the stated construction properties. This is a missing transfer, not a proof that no choices or relabeling could produce one. The reading covers the held mathematical reconstruction of the coloring rule and palette interface, not a new proof audit of the source report. No sublinear cube coloring follows from the inspected interface.

The remaining model questions are whether $\chi_r\ge cr$, whether $n_2(k)=O(k)$, and whether $\chi_8=5$. A family with $n/k\to\infty$ in $\mathbb F_2^n$ would give $\chi_n/n\to0$ along that family by the support map; subadditivity would then give $\chi_r=o(r)$ for all $r$. None of these questions is settled by the sparse-cell obstruction.