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arxiv: 2604.20106 · v1 · submitted 2026-04-22 · 🧮 math.CO

Monochromatic Sums and Quotients Near Zero

Md Moid Shaikh , Sourav Kanti Patra , Mukesh Kumar This is my paper

Pith reviewed 2026-05-10 00:44 UTC · model grok-4.3

classification 🧮 math.CO MSC 05D10
keywords partition regularitymonochromatic setsnear zerodense subsemigroupsadditive combinatoricscolorings of realsHindman theorem
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The pith

In any finite coloring of a dense subsemigroup of the positive reals, the set {a, b, ab, b(a+1)} is monochromatic for a and b arbitrarily close to zero.

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

The paper proves that for every finite coloring of a dense subsemigroup of ((0, ∞), +), there exist a and b in the subsemigroup, as small as one wishes, such that a, b, ab, and b(a+1) all receive the same color. It further shows that the pattern x, y, x+y, y/x is partition regular near zero. This extends earlier monochromatic results from the natural numbers to the reals close to the origin. A sympathetic reader would care because the configurations persist across scales, from discrete integers to dense continuous structures.

Core claim

We prove that {a, b, ab, b(a+1)} is monochromatic near zero, meaning that for every finite coloring of a dense subsemigroup of ((0, ∞), +), we obtain a, b in the subsemigroup as small as we want such that the set is monochromatic. We also show that the pattern x, y, x+y, y/x is partition regular near zero.

What carries the argument

The configuration {a, b, ab, b(a+1)} and its monochromaticity under finite colorings, together with the four-term pattern x, y, x+y, y/x, in the near-zero regime of dense additive subsemigroups.

If this is right

  • The configuration {a, b, ab, b(a+1)} remains unavoidable near zero in any finite coloring.
  • The pattern x, y, x+y, y/x is partition regular near zero in dense subsemigroups.
  • Variants of results on natural numbers, including certain Hindman-type statements, hold when restricted to arbitrarily small elements.
  • The disproof of Sahasrabudhe's conjecture on non-regularity extends to the near-zero setting.

Where Pith is reading between the lines

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

  • Other partition-regular patterns known on the integers may admit direct near-zero versions in the reals.
  • The result likely applies to other dense additive subsemigroups such as the positive rationals or algebraic positives.
  • One could test whether the same configurations remain monochromatic under different colorings or when the underlying operation changes.
  • Links to topological or measure-theoretic versions of Ramsey theory on the reals become plausible.

Load-bearing premise

The subsemigroup must be dense in the positives, closed under addition and the multiplications needed to produce ab and b(a+1), and contain arbitrarily small positive elements.

What would settle it

A 2-coloring of the positive rationals in which, for all sufficiently small a and b, the four numbers a, b, ab, and b(a+1) use both colors.

read the original abstract

Recently S. Goswami proved that whenever the set $\mathbb N$ of natural numbers is finitely colored, the set $\{a, b, ab, b(a+1)\}$ is monochromatic which also established a variant of the long-standing Hindman's conjecture, which asks for a monochromatic set of the form $\{a, b, ab, a+b\}$. Actually he disproved a conjecture proposed by J. Sahasrabudhe that $\{a, b, a(b + 1)\}$ is not partition regular. In this paper we prove that $\{a, b, ab, b(a+1)\}$ is monochromatic near zero which means for every finite coloring of a dense subsemigroups of $((0, \infty), +)$, the set $\{a, b, ab, b(a+1)\}$ is monochromatic near zero or in other words, we will get $a, b$ in a dense subsemigroups of $((0, \infty), +)$ as small as we want such that the set $\{a, b, ab, b(a+1)\}$ is monochromatic for every finite coloring of that dense subsemigroups of $((0, \infty), +)$, also we show that the pattern $x, y, x+y, \frac{y}{x}$ is partition regular near zero.

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 / 2 minor

Summary. The paper extends a result of Goswami on monochromatic configurations in finite colorings of the natural numbers to the setting of dense additive subsemigroups S of ((0, ∞), +). It claims that for any finite coloring of such an S, there exist arbitrarily small a, b ∈ S such that the set {a, b, ab, b(a+1)} is monochromatic. It further claims that the pattern x, y, x+y, y/x is partition regular near zero.

Significance. If the technical details are resolved, the results would provide a near-zero continuous analogue of Hindman-type theorems and partition regularity results, bridging discrete combinatorial number theory with dense subsets of the positive reals. The explicit construction of monochromatic configurations arbitrarily close to zero is a potentially useful strengthening.

major comments (2)
  1. [Abstract and §1] Abstract and §1 (statement of main theorems): The central claim requires that ab and b(a+1) lie in S whenever a, b ∈ S, so that they receive colors under the given coloring of S. However, S is defined only as a dense additive subsemigroup of ((0, ∞), +) with no stated closure under multiplication. The manuscript must either add an explicit hypothesis that S is closed under the relevant multiplications or prove that the chosen a, b can always be selected so that ab, b(a+1) ∈ S. This precondition is load-bearing for the well-posedness of the monochromatic configuration.
  2. [§2] §2 (proof of the first theorem): The argument that such a, b exist arbitrarily close to zero must explicitly verify that the constructed elements remain inside the given dense subsemigroup S after the multiplications are performed; otherwise the configuration exits the domain of the coloring.
minor comments (2)
  1. [Throughout] Notation for the dense subsemigroup S should be introduced once in §1 and used consistently; the repeated phrase 'dense subsemigroups of ((0, ∞), +)' is redundant.
  2. [Abstract and §3] The second claim on the pattern x, y, x+y, y/x requires closure under division; this should be stated explicitly alongside the additive-semigroup hypothesis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting these important points about the well-posedness of our configurations. We respond to each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1 (statement of main theorems): The central claim requires that ab and b(a+1) lie in S whenever a, b ∈ S, so that they receive colors under the given coloring of S. However, S is defined only as a dense additive subsemigroup of ((0, ∞), +) with no stated closure under multiplication. The manuscript must either add an explicit hypothesis that S is closed under the relevant multiplications or prove that the chosen a, b can always be selected so that ab, b(a+1) ∈ S. This precondition is load-bearing for the well-posedness of the monochromatic configuration.

    Authors: We agree that the current definition of S as a dense additive subsemigroup does not automatically ensure ab, b(a+1) ∈ S. The strongest honest revision is to add the explicit hypothesis that S is closed under multiplication (making S a dense subsemigroup of ((0,∞),+) that is also closed under the relevant products). This is the natural setting for configurations mixing sums and products, and it renders the monochromatic set well-defined within the colored domain. We will update the abstract, §1, and all theorem statements to include this hypothesis. We do not claim in the current manuscript that we already prove the products remain in S without it, so we will not attempt to add such a proof; the added hypothesis is the direct and accurate fix. revision: yes

  2. Referee: [§2] §2 (proof of the first theorem): The argument that such a, b exist arbitrarily close to zero must explicitly verify that the constructed elements remain inside the given dense subsemigroup S after the multiplications are performed; otherwise the configuration exits the domain of the coloring.

    Authors: We accept this criticism. In the revised §2 we will insert an explicit verification paragraph immediately after the construction of a and b, confirming that, under the new closure hypothesis, ab and b(a+1) belong to S. We will also make the choice of a and b more explicit so that the reader can see they remain inside S throughout the argument. This addresses the concern that the configuration could leave the domain of the coloring. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation self-contained

full rationale

The paper extends a cited result of Goswami on natural numbers to a statement about monochromatic configurations near zero in dense additive subsemigroups of (0,∞). The abstract and available text contain no self-definitional reductions, no fitted parameters renamed as predictions, no load-bearing self-citations, and no ansatzes smuggled via prior work by the same authors. The central claims rest on standard combinatorial partition-regularity arguments whose validity is independent of the present paper's inputs. No equation or step reduces by construction to a prior definition or fit within the manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard algebraic and combinatorial axioms with no free parameters or invented entities introduced in the abstract.

axioms (2)
  • standard math ((0, ∞), +) is a commutative semigroup
    Basic structure invoked for the ambient space and subsemigroups.
  • domain assumption Dense subsemigroups admit arbitrarily small positive elements closed under addition
    Required to locate the monochromatic configurations 'near zero'.

pith-pipeline@v0.9.0 · 5543 in / 1437 out tokens · 49287 ms · 2026-05-10T00:44:38.094148+00:00 · methodology

discussion (0)

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Reference graph

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