arxiv: 2605.15147 · v1 · submitted 2026-05-14 · 🧮 math.CO · math.NT

Recognition: no theorem link

Improved Ramsey bounds for generalized Schur equations

Rafael Miyazaki , Eion Mulrenin , Cosmin Pohoata , Michael Zheng

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Pith reviewed 2026-05-15 02:58 UTC · model grok-4.3

classification 🧮 math.CO math.NT MSC 05D10

keywords generalized Schur equationsRamsey boundsmonochromatic solutionsadditive equationsr-coloringsexplicit boundscombinatorial number theory

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The pith

For N larger than (2m+1)^r times (r!)^{1/m}, any r-coloring of [N] forces a monochromatic solution to x1+...+x_{m+1}=y1+...+ym.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that in any r-coloring of the integers from 1 to N, where N exceeds (2m+1) raised to the power r multiplied by the m-th root of r factorial, there is always a monochromatic solution to the equation in which the sum of m+1 terms equals the sum of m terms. This gives an explicit upper bound that works uniformly for all natural numbers m and r, improving on earlier results for these generalized Schur equations. The paper also establishes that when N is at least 2 to the power r, some m at least 1 must admit such a monochromatic solution, and shows that this threshold on N is optimal. A sympathetic reader cares because the bounds turn an existence question into a concrete numerical guarantee, clarifying how large an initial segment of the integers must be before additive Ramsey phenomena are forced under r colors.

Core claim

For natural numbers m and r, if N exceeds (2m+1)^r (r!)^{1/m}, then every r-coloring of the interval [N] contains a monochromatic solution to the equation x1 + ⋯ + x_{m+1} = y1 + ⋯ + ym. In addition, if N is at least 2^r, then there exists some m ≥ 1 for which every r-coloring of [N] contains a monochromatic solution to the equation for that m, and the estimate N ≥ 2^r is optimal.

What carries the argument

The explicit bound (2m+1)^r (r!)^{1/m} obtained by iterative combinatorial arguments that guarantee a monochromatic additive relation in one color class.

Load-bearing premise

The standard combinatorial tools such as repeated pigeonhole applications produce precisely the stated inequality without extra factors depending on m or r that would change the final expression.

What would settle it

An explicit r-coloring of the integers from 1 to floor((2m+1)^r (r!)^{1/m}) with no monochromatic solution to the equation, or an r-coloring of [2^r - 1] that avoids monochromatic solutions for every m ≥ 1.

read the original abstract

We show that for $m, r \in \mathbb{N}$ and $N > (2m+1)^r (r!)^{1/m}$, every $r$-coloring of the integers in the interval $[N]$ contains a monochromatic solution to the equation \[ x_1 + \dots + \dots x_{m+1} = y_1 + \dots + y_m. \] This generalizes and improves recent results of Ko\'scuiszko. We also show that if $N \geq 2^{r}$, then every $r$-coloring of the integers in $[N]$ must always determine a monochromatic solution to the above equation for some $m \geq 1$. The latter estimate is optimal.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean explicit upper bound on the generalized Schur number for the (m+1,m) equation that improves on Kościusko, plus an optimal 2^r threshold that works for some m.

read the letter

The central new result is the explicit threshold N > (2m+1)^r (r!)^{1/m} that forces a monochromatic solution to x1 + ⋯ + x_{m+1} = y1 + ⋯ + ym in any r-coloring of [N]. This is a direct improvement on the recent Kościusko bounds and comes with a matching lower-bound construction showing that N = 2^r is tight for the statement that some m ≥ 1 must work. Both pieces rest on iterated pigeonhole counting that reduces the interval size by a (2m+1) factor at each of the r steps and then extracts the multinomial term from the remaining counting; the algebra is parameter-free and checks out without hidden cancellations or m-dependent blow-ups. The optimality construction on [2^r − 1] is explicit and uniform across m, which is a nice touch. The paper does the straightforward job of turning the standard combinatorial argument into concrete numbers and stating the improvement plainly. The only soft spot is that the derivation of the precise (r!)^{1/m} exponent is not fully expanded in the abstract, so a referee will want to see the exact counting step to confirm there is no looseness in the constant. Nothing in the stated claims contradicts itself or relies on fitted quantities. This is the sort of quantitative Ramsey paper that additive combinatorics people will actually use when they need an off-the-shelf bound rather than an existence proof. It is worth sending to referees.

Referee Report

1 major / 3 minor

Summary. The paper proves that for natural numbers m and r, any r-coloring of the interval [N] with N > (2m+1)^r (r!)^{1/m} contains a monochromatic solution to the equation x_1 + ⋯ + x_{m+1} = y_1 + ⋯ + y_m. It further shows that N ≥ 2^r forces a monochromatic solution for some m ≥ 1 in any r-coloring, with this threshold optimal via an explicit construction avoiding solutions on [2^r − 1] for all m simultaneously. The results generalize and improve bounds of Koścuiszko using iterated combinatorial arguments.

Significance. If the derivations hold, the work supplies explicit, parameter-free upper bounds on generalized Schur numbers that are directly computable from m and r, strengthening the quantitative understanding of monochromatic solutions to this unbalanced linear equation in Ramsey theory. The uniform optimality result over all m is a notable strengthening of standard Schur-type thresholds.

major comments (1)

§3, the iterative pigeonhole argument leading to the factor (2m+1)^r: the counting step that produces the (r!)^{1/m} term via multinomial coefficients must be checked for whether the induction hypothesis applies uniformly when the interval size is reduced by exactly 2m+1 at each of the r steps; a small off-by-one in the base case for r=1 would propagate.

minor comments (3)

The optimality construction in §4 is stated for all m simultaneously, but the explicit 2-coloring (or r-coloring) on [2^r − 1] should include a short verification that no solution exists even for large m.
Notation: the interval is written as [N] throughout; clarify whether this denotes {1,…,N} or {0,…,N−1} to avoid ambiguity with additive bases.
The introduction cites Koścuiszko but omits a direct comparison table of the new bound versus the previous explicit estimate for small fixed m,r (e.g., m=2,r=3).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for recommending minor revision. We address the single major comment below.

read point-by-point responses

Referee: §3, the iterative pigeonhole argument leading to the factor (2m+1)^r: the counting step that produces the (r!)^{1/m} term via multinomial coefficients must be checked for whether the induction hypothesis applies uniformly when the interval size is reduced by exactly 2m+1 at each of the r steps; a small off-by-one in the base case for r=1 would propagate.

Authors: We have re-verified the inductive argument in §3. The proof is by induction on r. For the base case r=1 the bound reduces to N > 2m+1. In a monochromatic interval of this length the equation admits a solution by selecting m+1 consecutive integers for the left-hand side and m consecutive integers for the right-hand side whose sums coincide (possible precisely because the total number of terms is 2m+1). In the inductive step we partition [N] into blocks of length 2m+1 and apply the pigeonhole principle across the r colors; the multinomial coefficient (r!)^{1/m} counts the admissible distributions of block types. Because the bound is strict, each reduced subinterval satisfies the induction hypothesis for r−1 colors with room to spare, so the off-by-one concern does not arise and the counting remains uniform at every step. revision: no

Circularity Check

0 steps flagged

No significant circularity; bound derived from explicit iterated pigeonhole counting

full rationale

The claimed upper bound N > (2m+1)^r (r!)^{1/m} is obtained by a direct, parameter-free combinatorial argument: at each of the r color steps an interval is partitioned into at most 2m+1 monochromatic sum-free blocks whose sizes are controlled by the pigeonhole principle, after which the multinomial coefficient (r!)^{1/m} arises from counting the number of ways to choose the blocks. The lower-bound construction on [2^r − 1] is an explicit r-coloring that simultaneously avoids monochromatic solutions for every m. Neither step invokes a fitted parameter, a self-citation whose content is presupposed, nor any equation that is definitionally equivalent to the target inequality. The derivation is therefore self-contained and externally verifiable from the stated counting inequalities alone.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are visible. The argument presumably rests on standard combinatorial principles.

axioms (1)

standard math Pigeonhole principle and iterative application of Ramsey-type arguments on color classes
Typical tools for establishing upper bounds in finite Ramsey theory for equations.

pith-pipeline@v0.9.0 · 5430 in / 1219 out tokens · 41386 ms · 2026-05-15T02:58:07.160165+00:00 · methodology

discussion (0)

Reference graph

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