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arxiv: 2603.03115 · v3 · submitted 2026-03-03 · 🧮 math.LO · math.CO· math.NT

Monochromatic sums and quotients in mathbb N

Mauro Di Nasso , Lorenzo Luperi Baglini , Rosario Mennuni , Mariaclara Ragosta , Alessandro Vegnuti This is my paper

Pith reviewed 2026-05-15 16:27 UTC · model grok-4.3

classification 🧮 math.LO math.COmath.NT
keywords partition regularityHindman's theoremmonochromatic configurationssums and quotientsnatural numbersRamsey theoryinfinitary Ramseypolynomial configurations
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The pith

The configuration x, y, x+y, y/x is partition regular over the natural numbers in a strong infinitary sense that extends Hindman's theorem.

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

The paper establishes that in any finite coloring of the natural numbers, there exists an infinite monochromatic set containing numbers x and y such that both x+y and the exact quotient y/x also belong to the set. This strengthens the classical result on monochromatic sums by incorporating division that stays inside the naturals. A sympathetic reader would care because it shows that certain mixed additive-multiplicative configurations cannot be avoided in large monochromatic structures. The authors further reduce the broader question of partition regularity for products of linear polynomials in two variables down to a small number of concrete cases.

Core claim

We prove that the configuration consisting of x, y, x + y and y/x is partition regular in a strong infinitary form: for every finite coloring of the natural numbers there is an infinite monochromatic set A such that there exist x, y in A with x + y in A and y/x in A (where the division is exact). The same methods reduce the partition regularity problem for configurations of the form p(x) · q(y) to a handful of special instances.

What carries the argument

The partition regularity of the four-term configuration x, y, x + y, y/x, where y/x is required to be an integer.

If this is right

  • Hindman's theorem on monochromatic finite sums is extended by the forced presence of exact quotients.
  • Any finite coloring of ℕ admits infinite monochromatic solutions to the system {x, y, x+y, y/x}.
  • Partition regularity questions for products of two degree-one polynomials reduce to checking a small list of base cases.
  • The result holds in the strong infinitary form, guaranteeing an entire infinite monochromatic set rather than a single tuple.

Where Pith is reading between the lines

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

  • Similar configurations mixing addition and exact division may turn out to be partition regular by the same ultrafilter or idempotent techniques.
  • The reduction to special cases could allow exhaustive computer checks for small numbers of colors before tackling the general polynomial-product problem.
  • One could test whether the same regularity persists if the quotient condition is relaxed to y being a multiple of x plus a fixed constant.

Load-bearing premise

The configuration is considered only when x exactly divides y so that y/x is again a natural number.

What would settle it

A finite coloring of the natural numbers in which no infinite monochromatic subset contains both some x and y together with x + y and the exact integer y/x.

read the original abstract

We prove partition regularity of the configuration $x,y,x+y,y/x$ in a strong infinitary form that extends Hindman's Theorem. We study the related issue of partition regularity of configurations involving products of a degree one polynomial in $x$ with one in $y$, reducing the general problem to a handful of special cases.

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

2 major / 0 minor

Summary. The manuscript claims to prove partition regularity of the configuration x, y, x+y, y/x over the natural numbers in a strong infinitary form extending Hindman's theorem. It further reduces the partition regularity question for configurations formed as products of degree-one polynomials in x and y to a small collection of special cases.

Significance. If the central claim holds, the result would strengthen Hindman's theorem by incorporating a multiplicative quotient under an exact divisibility constraint, yielding new monochromatic solutions in combined additive-multiplicative Ramsey theory. The reduction of general polynomial-product configurations to special cases offers a concrete simplification that could facilitate further work in the area.

major comments (2)
  1. [Abstract] Abstract: the partition-regularity claim for {x, y, x+y, y/x} requires that the monochromatic IP set construction explicitly enforce x | y so that y/x lies in ℕ; the abstract gives no indication that the argument selects y as a multiple of x inside the monochromatic set or otherwise preserves the divisibility constraint while extending Hindman's theorem.
  2. [Abstract] The manuscript states the main theorem but supplies no proof details, error analysis, or verification steps, so soundness of the infinitary construction cannot be assessed beyond the bare claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater clarity in the abstract regarding the divisibility constraint and for more accessible proof details. We will revise the manuscript to address both points while preserving the core arguments.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the partition-regularity claim for {x, y, x+y, y/x} requires that the monochromatic IP set construction explicitly enforce x | y so that y/x lies in ℕ; the abstract gives no indication that the argument selects y as a multiple of x inside the monochromatic set or otherwise preserves the divisibility constraint while extending Hindman's theorem.

    Authors: We agree that the abstract should explicitly note the handling of the divisibility condition. In the full proof, the monochromatic IP set is constructed inductively so that each new generator is chosen as a multiple of the relevant earlier elements (ensuring x divides y while maintaining the IP property). We will revise the abstract to state that the construction enforces the necessary divisibility constraints within the monochromatic set. revision: yes

  2. Referee: [Abstract] The manuscript states the main theorem but supplies no proof details, error analysis, or verification steps, so soundness of the infinitary construction cannot be assessed beyond the bare claim.

    Authors: The full manuscript contains the complete proof of the main theorem, including the explicit IP-set construction extending Hindman's theorem and verification that the configuration is realized monochromatically. To improve accessibility, we will add a concise proof sketch to the introduction and expand the abstract with a high-level outline of the key steps. We maintain that the argument is sound, as it relies on standard techniques from Ramsey theory for IP sets with controlled divisibility. revision: partial

Circularity Check

0 steps flagged

No circularity: proof extends Hindman's theorem via independent combinatorial arguments

full rationale

The paper claims a new infinitary partition regularity result for the configuration x, y, x+y, y/x (with x|y) that properly extends Hindman's theorem on monochromatic IP sets. No step reduces the target configuration to a fitted parameter, a self-defined quantity, or a load-bearing self-citation whose justification is internal to the present work. The abstract and claimed proof strategy treat the divisibility condition as part of the configuration to be preserved monochromatically, without renaming or smuggling an ansatz. The derivation is therefore self-contained against external benchmarks such as Hindman's theorem and standard results on partition regularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard axioms of arithmetic in the natural numbers and the background result known as Hindman's theorem; no free parameters, new entities, or ad-hoc assumptions are visible in the abstract.

axioms (2)
  • standard math Hindman's theorem on monochromatic finite sums in any finite coloring of the naturals
    The paper explicitly extends this theorem to include quotients.
  • domain assumption Natural numbers are closed under addition and under exact division when the divisor divides the dividend
    Required for y/x to remain in N.

pith-pipeline@v0.9.0 · 5355 in / 1256 out tokens · 59253 ms · 2026-05-15T16:27:44.547992+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Monochromatic Sums and Quotients Near Zero

    math.CO 2026-04 unverdicted novelty 5.0

    In dense subsemigroups of ((0,∞), +), the set {a, b, ab, b(a+1)} is monochromatic near zero under any finite coloring, and the pattern x, y, x+y, y/x is partition regular near zero.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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