Exponential or superexponential growth of the Schur numbers

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This lead records a route by Plasma AI to Problem 483: is there an absolute $C$ with $f(k)<C^k$, where $f(k)=S(k)+1$ and $S(k)$ is the largest $N$ such that $[1,N]$ has a coloring by $k$ colors with no monochromatic $x+y=z$, equal summands allowed? A full answer is either such a $C$ or a family of colorings with unbounded $(N+1)^{1/k}$. The problem page's Current assessment records the status as open, with growth rate at least $380^{1/5}$ and the factorial upper bound $(e-\tfrac16)k!$.

The canonical argument is the proof sketch, with its companions in the Schur-growth notes. All results there are author-recorded at tier 0 or open; none has been independently examined.

Route

Both directions pass through translation-invariant colorings. The upper direction needs a color-elimination step that loses only an absolute factor per removed color. Three uniform forms of it are refuted (L260, L261, L266), and the signed-cube bound L264 would suffice through L263. The lower direction needs a mechanism that, unlike the bound $f(k)\le R_k(3)-1$, produces colorings depending only on differences. The mechanism behind the accepted superexponential bound for $R_k(3)$ is a palette rule. The palette-rule page proves that under translation invariance it gains at most a factor $2$ per color on every partition into cosets of a finite abelian group or into subcubes of the signed cube, mixed partitions included, and on residue-class partitions of an interval up to one color per block, where its recursions stay exponential. That transfer is therefore closed in these settings, with its failed input identified as false.

The two-sided replacement with fixed binary linear quotient maps is also capped. Proposition 8 on that page uses Brouwer's binary affine-cover theorem to remove $t$ colors at a cost of at most $t$ dimensions while preserving the model. Thus a zero-seed structure has dimension at most its color count, regardless of label ranks or the number of stages. The proof also permits changing kernels under condition (A) on that page: the new layer lies in one nonzero coset whose direction avoids the earlier class. This closes the specified replacements, not general sum-free colorings of vector spaces or intervals.

Open tests

• Two-sided reuse through fibres outside condition (A). A reopening example must cover every coarse coset and satisfy both two-block and three-block constraints on the palette-rule page, with a measured gain in size per color. A cyclic-group example must also control repeated summands; a binary example alone supplies no Schur transfer. No such construction or preserved Bohr-label invariant is supplied. A discriminating example would have actual fibre differences avoiding the earlier class while their linear span meets that class; replacement by a whole coset would then fail.
• Intervals: palette recursions on residue-class partitions are proved exponential; partitions into progressions that are not residue classes are open.
• Cube blocks that are not subcubes: a retained witness shows that the palette bound fails for twisted blocks. The witness factors through neither binary values nor supports, so a gain there still needs an interval transfer.
• On the upper side, the extremal prefix assertion and an adaptive Bohr recursion, both recorded in the sketch as open reformulations without a supporting invariant.

The route is blocked on a new construction or invariant: the specified palette and fixed-quotient mechanisms are capped, no tested family beats $380^{1/5}$, and the upper recursion still lacks an absolute loss per removed color. The broader growth question remains open. Increasing the number of stages or the ranks of the same linear labels cannot reopen the fixed-quotient model.

Current review

No independent review of this route is recorded.