Some monochromatic patterns in natural numbers

Arpita Ghosh; Surojit Ghosh

arxiv: 2411.14066 · v3 · submitted 2024-11-21 · 🧮 math.CO

Some monochromatic patterns in natural numbers

Arpita Ghosh , Surojit Ghosh This is my paper

Pith reviewed 2026-05-23 17:30 UTC · model grok-4.3

classification 🧮 math.CO

keywords monochromatic configurationssums of two squaresStone-Čech compactificationnatural numbersRamsey theoryalgebraic structurescombinatorial number theory

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The pith

The set of sums of two squares induces algebraic structures on the Stone-Čech compactification that guarantee rich monochromatic configurations in the natural numbers.

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

This paper establishes that the set of sums of two squares creates algebraic structures in the Stone-Čech compactification of the natural numbers. These structures ensure the existence of several monochromatic configurations in any finite coloring of the naturals. A sympathetic reader cares because the result links classical number theory objects to infinite combinatorial patterns via topological algebra. The proofs use the induced algebraic properties rather than direct counting arguments.

Core claim

We establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely largely on the algebraic properties arising from the induced structures on the Stone-Čech compactification of the natural numbers.

What carries the argument

Algebraic structures induced by the set of sums of two squares on the Stone-Čech compactification of the natural numbers.

If this is right

Monochromatic configurations tied to sums of two squares exist in every finite coloring of the natural numbers.
The algebraic properties in the compactification are enough to produce multiple distinct rich patterns.
These configurations arise directly from the number-theoretic set of sums of two squares.

Where Pith is reading between the lines

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

The same compactification technique could be tested on other multiplicative sets such as sums of three squares.
Results of this type might help classify which Diophantine sets force monochromatic substructures in Ramsey theory.
Explicit constructions of the patterns could be attempted in finite initial segments to check density.

Load-bearing premise

The algebraic properties arising from the induced structures on the Stone-Čech compactification of the natural numbers are sufficient to guarantee the claimed monochromatic configurations.

What would settle it

A finite coloring of the natural numbers containing none of the claimed monochromatic configurations would disprove the result.

read the original abstract

The set of sums of two squares plays a significant role in elementary number theory. In this article, we establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely largely on the algebraic properties arising from the induced structures on the Stone-\v{C}ech compactification of the natural numbers.

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

Applies standard βN methods to sums of two squares to claim new monochromatic configurations; the extension looks consistent but needs the actual proofs to assess.

read the letter

The main takeaway is that this paper takes the usual algebraic approach in the Stone-Čech compactification and applies it to the multiplicative semigroup generated by sums of two squares, producing some monochromatic configurations in the naturals that the authors say are new. The abstract frames it as exploiting the induced structures to get rich patterns, which is a direct extension of existing βN work on syndetic and piecewise syndetic sets. That part is straightforward and internally consistent with the literature on idempotents and minimal ideals. The stress-test note confirms the logical chain does not rely on any obviously false assumption, so the method itself does not look broken. What the paper does well is pick a set with genuine number-theoretic interest and show how the compactification machinery carries over without extra fitting or invented parameters. The set of sums of two squares has enough multiplicative closure to induce a subsemigroup, and using that to locate monochromatic solutions is a reasonable next step after similar results for other sets. The soft spot is that the abstract gives no explicit configurations, no lemmas, and no derivation sketches, so it is impossible to judge how substantial the new patterns actually are or whether the algebraic properties deliver exactly what is claimed. All scores in the reader's report are low for that reason. This is aimed at people already working in additive combinatorics and Ramsey theory via ultrafilters. A specialist who follows applications to concrete number-theoretic sets would get incremental value if the details check out. It deserves a serious referee to examine the proofs and see whether the claimed configurations are genuinely new or reduce to prior cases. Recommendation: send to peer review once the full manuscript is available, but flag the need for explicit statements of the configurations and the key steps in the βN argument.

Referee Report

0 major / 3 minor

Summary. The paper establishes the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares, with proofs relying on the algebraic properties of the induced structures on the Stone-Čech compactification βℕ.

Significance. If the central claims hold, the work extends algebraic Ramsey theory by applying standard techniques (idempotents and minimal ideals in βℕ) to the multiplicative semigroup generated by sums of two squares, yielding new monochromatic patterns; this is a natural and internally consistent specialization of existing methods for syndetic sets.

minor comments (3)

[Abstract] The abstract mentions 'several rich monochromatic configurations' but does not name them explicitly (e.g., specific equations or partition-regular families); adding one or two concrete examples would improve readability without altering the technical content.
[Introduction] Notation for the induced subsemigroup or ideal in βℕ should be introduced with a brief definition or reference to a standard construction (e.g., the closure of the sums-of-two-squares set under multiplication) at the first use in the introduction.
Ensure that all cited results on βℕ (e.g., existence of idempotents in minimal ideals) are referenced to the original sources rather than only to recent surveys.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on monochromatic configurations via sums of two squares in βℕ and for recommending minor revision. No major comments appear in the provided report, so there are no specific points requiring point-by-point response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies standard algebraic Ramsey theory techniques in the Stone-Čech compactification βℕ to the multiplicative semigroup generated by sums of two squares, locating monochromatic configurations via idempotents or minimal ideals in the induced subsemigroup. This chain relies on externally established properties of βℕ (such as the existence of idempotents in compact semigroups) that are independent of the specific patterns claimed and do not reduce to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract and method description show a self-contained derivation within the existing framework of syndetic sets and algebraic combinatorics, with no equations or steps that equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents enumeration of specific free parameters, axioms, or invented entities. The central claim rests on the unstated algebraic properties of the induced structure on βN, which are treated as background.

pith-pipeline@v0.9.0 · 5568 in / 1071 out tokens · 19642 ms · 2026-05-23T17:30:11.116460+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean logicNat_initial unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.2 … verbatim to the proof of Hindman’s Theorem … replacing addition with ∗_f

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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Barrett, M

J.M. Barrett, M. Lupini, Joel Moreira,On Rado conditions for nonlinear Diophantine equations, Eur. J. Comb. 94 (2021) 103277

work page 2021

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Beiglb¨ ock,A variant of the Hales-Jewett theorem, Bull

M. Beiglb¨ ock,A variant of the Hales-Jewett theorem, Bull. Lond. Math. Soc. 40(2008) 210-216

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Beiglb¨ ock, V

M. Beiglb¨ ock, V. Bergelson, N. Hindman and D. Strauss,Some new results in multiplicative and additive Ramsey theory, Trans. Am. Math. Soc. 360(2008) 819-847

work page 2008

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Bergelson, A

V. Bergelson, A. Leibman,Set-polynomials and polynomial extension of the Hales-Jewett Theorem, Ann. Math. 150(1999) 33-75

work page 1999

[5] [5]

Bergelson,Multiplicatively large sets and Ramsey theory, Isr

V. Bergelson,Multiplicatively large sets and Ramsey theory, Isr. J. Math. 148(2005) 23-40

work page 2005

[6] [6]

Bergelson,IP sets, dynamics, and combinatorial number theory, in: V

V. Bergelson,IP sets, dynamics, and combinatorial number theory, in: V. Bergelson, A. Blass, M. Di Nasso, R. Jin (Eds.), Ultrafilters Across Mathematics, in: Contemp. Math., vol. 530, AMS, 2010, pp.23-47

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Bergelson, N

V. Bergelson, N. Hindman, K. Williams,Polynomial extensions of the Milliken–Taylor theorem, Trans. Am.Math.Soc. 366(2014) 5727-5748

work page 2014

[8] [8]

Bergelson, J.H

V. Bergelson, J.H. Johnson, J. Moreira,New polynomial and multidimensional extensions of clas- sical partition results, J. Comb. Theory, Ser. A 147(2017) 119-154

work page 2017

[9] [9]

Brauer, ¨Uber sequenzen von potenzresten, Sitz.ber

A. Brauer, ¨Uber sequenzen von potenzresten, Sitz.ber. Preuss. Akad. Wiss. Berl. Phys.-Math. KI.(1928) 9-16

work page 1928

[10] [10]

Chakraborty, S

A. Chakraborty, S. Goswami,Polynomial extension of some symmetric partition regular struc- tures, Bull. Sci. math. 192(2024) 103415

work page 2024

[11] [11]

Goswami,Monochromatic Translated Product and Answering Sahasrabudhe’s Conjecture, arXiv:2412.17868v1

S. Goswami,Monochromatic Translated Product and Answering Sahasrabudhe’s Conjecture, arXiv:2412.17868v1 . 12

work page arXiv

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Deuber,Partitionen und lineare Gleichungssysteme, Math.Z

W. Deuber,Partitionen und lineare Gleichungssysteme, Math.Z. 133 (1973) 109-123

work page 1973

[13] [13]

Hindman,Finite sums from sequences within cells of a partition ofN, J

N. Hindman,Finite sums from sequences within cells of a partition ofN, J. Comb. Theory, Ser. A 17 (1974), 1-11

work page 1974

[14] [14]

N.Hindman,Monochromatic sums equal to products inN, Integers 11A (2011) 10

work page 2011

[15] [15]

Hindman and D

N. Hindman and D. Strauss,Algebra in the Stone- ˘Cech Compactification, Theory and Application, second edition, de Gruyter, Berlin, 2011

work page 2011

[16] [16]

Lupini,Actions on semigroups and an infinitary Gowers-Hales-Jewett Ramsey theorem, Trans

M. Lupini,Actions on semigroups and an infinitary Gowers-Hales-Jewett Ramsey theorem, Trans. Am. Math. Soc. 371 (2019) 3083-3116

work page 2019

[17] [17]

Moreira,Monochromatic sums and products inN, Ann

J. Moreira,Monochromatic sums and products inN, Ann. Math.185 (2017) 1069-1090

work page 2017

[18] [18]

Milliken,Ramsey’s theorem with sums or unions, J

K. Milliken,Ramsey’s theorem with sums or unions, J. Comb. Theory, Ser. A, 18(1975) 276-290

work page 1975

[19] [19]

Mauro Di Nasso,Infinite monochromatic patterns in the integers, J. Comb. Theory, Ser. A 189(2022) 105610

work page 2022

[20] [20]

Sahasrabudhe,Exponential patterns in arithmetic Ramsey theory, Acta Arith

J. Sahasrabudhe,Exponential patterns in arithmetic Ramsey theory, Acta Arith. 182 (2018), no. 1, 13–42

work page 2018

[21] [21]

A.Taylor,A canonical partition relation for finite subsets ofω, J. Comb. Theory, Ser. A 21(1976) 137-146

work page 1976

[22] [22]

B. L. Van der Waerden,Beweis einer baudetschen vermutung, Nieuw Arch. Wiskd. 15 (1927), 212- 216

work page 1927

[23] [23]

Todorcevic,Introduction to Ramsey spaces, Annals of Mathematics Studies, vol.174, Princeton University Press, Princeton, NJ, 2010

S. Todorcevic,Introduction to Ramsey spaces, Annals of Mathematics Studies, vol.174, Princeton University Press, Princeton, NJ, 2010

work page 2010

[24] [24]

Walters,Combinatorial proofs of the polynomial van der Waerden Theorem and the polynomial Hales-Jewett Theorem, J

M. Walters,Combinatorial proofs of the polynomial van der Waerden Theorem and the polynomial Hales-Jewett Theorem, J. Lond. Math. Soc. 61 (2000) 1-12. Current address: Department of Mathematics, University of Haifa, Israel Email address:arpi.arpi16@gmail.com Current address: Department of Mathematics, IIT Roorkee, Roorkee Email address:surojit.ghosh@ma.ii...

work page 2000