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arxiv: 2604.23738 · v1 · submitted 2026-04-26 · 🧮 math.CO

Modular Schur numbers

Tom Sanders This is my paper

Pith reviewed 2026-05-08 05:48 UTC · model grok-4.3

classification 🧮 math.CO
keywords modular Schur numberslinear equationsRamsey theoryadditive combinatoricscolorings of Z/mZSchur numbers
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The pith

Modular Schur numbers for linear equation systems depend only on the number of equations, independent of their coefficients.

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

The paper examines modular analogues of Schur numbers that capture the smallest threshold guaranteeing a monochromatic solution to a system of linear equations inside the cyclic group Z/mZ. It proves that this threshold is determined solely by the number of equations in the system and does not change when the coefficients are altered. For systems consisting of exactly one equation the work also supplies improved bounds that are stronger than those known for the ordinary Schur numbers.

Core claim

The modular Schur number attached to any system of k linear equations over Z/mZ is the same for every choice of coefficients. When k=1 this common value satisfies stronger upper and lower bounds than the classical Schur number.

What carries the argument

The modular Schur number of a system of linear equations, proved invariant under arbitrary coefficient changes once the number of equations is fixed.

If this is right

  • All single linear equations, such as x+y=z and 2x=y, share the same modular Schur number for each m and r.
  • Improved bounds are available for every one-equation modular Schur number.
  • The modular Schur number for k-equation systems grows with k but remains fixed once k is chosen.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The invariance suggests that the Ramsey-theoretic behavior of linear equations in finite rings is governed by the number of independent constraints rather than by the specific arithmetic relations.
  • The same independence may hold for other additive Ramsey numbers or for equations over rings that are not cyclic.

Load-bearing premise

The avoidance of monochromatic solutions to linear equations in colorings of Z/mZ depends only on the number of equations and holds uniformly for any choice of coefficients and any modulus.

What would settle it

Exhibiting a modulus m, a number of colors r, and two single linear equations over Z/mZ whose modular Schur numbers differ would refute the claim.

Figures

Figures reproduced from arXiv: 2604.23738 by Tom Sanders.

Figure 1
Figure 1. Figure 1: Black squares represent sets in P; red rectangles represent sets in Q. view of the sizes of Q and A, by the Hales-Jewett Theorem (Theorem 2.7) there is z ∈ AQ and ∅ ̸= W ⊂ Q such that L := {x ∈ A Q : ∃a ∈ A with x|W ≡ a and x|Q\W = z|Q\W} is monochromatic. Let E := {Z} ∪ {S1, . . . , Sd : S ∈ Q \ W}, and P ′ := {P1, . . . , Pd} where Pi = [ S∈W Si . Since W is non-empty the set Pi is non-empty, and by desi… view at source ↗
read the original abstract

We study modular analogues of Schur numbers for systems of linear equations. We show that these only depend on the number of equations, not their coefficients and in the case of one equation show stronger bounds.

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.

Referee Report

1 major / 1 minor

Summary. The paper defines modular Schur numbers associated to systems of linear equations over Z/mZ and claims to prove that these numbers depend only on the number k of equations in the system, independent of the specific integer coefficients; for the special case k=1 it derives stronger explicit bounds.

Significance. If the independence result holds, it would simplify the landscape of modular Ramsey numbers by reducing all k-equation systems to a single parameter, allowing uniform bounds and constructions across coefficient choices. The stronger k=1 bounds would supply concrete, usable estimates in additive combinatorics.

major comments (1)
  1. [Abstract] The central claim (abstract and introduction) asserts coefficient-independence for arbitrary systems, yet the stress-test concern about degenerate cases (coefficients divisible by m, rank drop, or identically zero equations) is not addressed in the provided text. Without an explicit case split or reduction showing that both the lower-bound coloring and the upper-bound forcing argument remain uniform when the system degenerates, the independence statement lacks support and is load-bearing for the main theorem.
minor comments (1)
  1. The abstract supplies no definition of the modular Schur number, no proof sketch, and no statement of the precise range of m or k considered; expanding it would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to address degenerate cases explicitly. We respond to the major comment below and will update the manuscript to strengthen the presentation of the main result.

read point-by-point responses

  1. Referee: [Abstract] The central claim (abstract and introduction) asserts coefficient-independence for arbitrary systems, yet the stress-test concern about degenerate cases (coefficients divisible by m, rank drop, or identically zero equations) is not addressed in the provided text. Without an explicit case split or reduction showing that both the lower-bound coloring and the upper-bound forcing argument remain uniform when the system degenerates, the independence statement lacks support and is load-bearing for the main theorem.

    Authors: We agree that the manuscript does not contain an explicit discussion or case analysis of degenerate systems. The central claim as currently stated applies to arbitrary systems, but the proofs of the lower and upper bounds are written in a form that assumes the equations are presented as given without separately verifying the zero-coefficient or rank-deficient situations. In the revised manuscript we will add a short subsection in the preliminaries that treats these cases directly: when one or more equations become identically zero modulo m, the system reduces to one with strictly fewer than k equations; the modular Schur number for the original system is then at most the number for the reduced system. Because our main theorem already shows that the number depends only on the number of equations, this reduction is consistent once the statement is restricted to non-degenerate systems (i.e., systems in which no equation is the zero equation). We will also insert a brief remark after the statement of the main theorem noting that the independence result is henceforth understood to apply to non-degenerate systems of exactly k equations. The existing coloring construction and the iterated pigeonhole argument require no alteration, as they continue to supply valid bounds once the scope is clarified. These changes will be incorporated in the next version. revision: yes

Circularity Check

0 steps flagged

Direct combinatorial theorem with no reduction to inputs or self-citations

full rationale

The paper states a theorem that modular Schur numbers for k-equation systems over Z/mZ depend only on k and not on the specific coefficients. This is presented as a direct result without any fitted parameters, empirical predictions, or load-bearing self-citations. The abstract and claim contain no equations that define the quantity in terms of itself, no renaming of known patterns, and no ansatz smuggled via prior work. The derivation chain is therefore self-contained as a standard proof in combinatorial number theory, with no step reducing by construction to the input statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; no explicit free parameters, ad-hoc axioms, or new entities are introduced in the provided text. The result appears to rest on standard definitions from additive combinatorics.

axioms (1)
  • standard math Standard properties of linear equations and colorings over the integers modulo m
    The modular setting presupposes basic facts about Z/mZ that are not proved in the abstract.

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