Homogeneous Patterns in Ramsey Theory

Sayan Goswami; Sukumar Das Adhikari

REVIEW 6 major objections 4 minor 33 references

Homogeneous Patterns in Ramsey Theory

T0 review · 6 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Nonlinear equations can be built with any prescribed degree of regularity.

desk verdict The paper's new statements are interesting, but the main proofs rely on a false rational-polynomial van der Waerden theorem; recommend peer review with expectation of major revision. read the letter →

arxiv 2501.17203 v2 pith:AKZF6BBG submitted 2025-01-28 math.CO

Sukumar Das Adhikari , Sayan Goswami This is my paper · ORCID

classification math.CO MSC 05D1005C5522A1554D35

keywords RamseytheorypartitionregularityhomogeneouspatternspolynomialvanderWaerdentheoremPythagoreantriplesdegreeofp-adicvaluations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims three nonlinear Ramsey-theoretic results from one homogeneous-pattern framework. First, every finite coloring of the positive integers contains an infinite set $A$ and an arbitrarily large finite set $B$ such that $A \cup (A+B) \cup (A \cdot B)$ is monochromatic, settling the finitary version of a question posed in [27]. Second, for every positive integer $n$ and rational $c$, the equations $X^2+Y^2=Z^2+cU^nV$ and $X^2+Y^2=Z^2+c(U\pm V)$ are 2-regular, meaning every 2-coloring has a monochromatic solution. Third, for every $m$ and $n$ there is an $m$-degree homogeneous equation that is $n$-regular but not $(n+1)$-regular, giving a nonlinear analogue of the classical degree-of-regularity conjecture. If correct, the results transfer any future improvement on Pythagorean-triple regularity to the perturbed equations and realize every desired degree of regularity in every degree.

What carries the argument

The proof runs on three tools. The first is a rational-polynomial extension of the polynomial van der Waerden theorem (Theorem 2.1): for any finite set $\mathcal{F}$ of rational polynomials with zero constant term and any finite coloring, some color class contains infinitely many $x$ for which $x$, $xy$, and $x+P(y)$ all lie in that color for a fixed $y$. The second is the notion of homogeneous families of subsets of $\mathbb{Z}^+$, families closed under multiplication by positive integers, used to lift a single monochromatic configuration to the scaled copies needed for the Pythagorean-triple and degree-of-regularity constructions. The third is the $p$-adic valuation $\mathrm{ord}_p$, whose divisibility-additivity underlies the contradiction that proves non-regularity in Theorem 1.5.

What would settle it

Take the smallest nontrivial case of Theorem 2.1, $\mathcal{F}=\{P(x)=x/2\}$, and computationally search 2-colorings of $[1,N]$ for a fixed $y$ and a color class containing infinitely many $x$ with $x$, $xy$, and $x+y/2$ all integer and monochromatic; if a coloring persists with no valid $y$, the rational-polynomial step is false. Alternatively, for Theorem 1.5, a bounded search for a monochromatic solution of $M_n$ under the coloring $r \mapsto \mathrm{ord}_p(r) \bmod n$ would directly test the claimed non-$n$-regularity.

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Extended reading notes

Core claim

The central discovery is Theorem 1.5. For any prime $p$ and positive integers $m,n$ with $\gcd(m,n)=1$, the homogeneous equation $$\sum_{i=1}^{n-1} $p^{{mi}}$ X_{i,1}X_{i,2}^{m-1}=$X_n^{{m-1}}$X_{n+1}$$ is $(n-1)$-regular but not $n$-regular. Reindexing $n$ to $n+1$ gives an $m$-degree homogeneous equation that is $n$-regular but not $(n+1)$-regular when the coprimality hypothesis holds, which the paper presents as a nonlinear analogue of the degree-of-regularity conjecture. The non-regularity direction is proved by coloring each integer $r$ by $\mathrm{ord}_p(r)$ modulo $n$: under a hypothetical monochromatic solution, the coefficients $p^{mi}$ force all terms to have distinct $p$-adic valuations, contradicting that their sum is zero. The $m=1$ case is linear and recovers the known resolution of the classical degree-of-regularity conjecture.

Load-bearing premise

Everything rests on the claim that the polynomial van der Waerden theorem's conclusion remains valid when the polynomials are allowed to have rational coefficients rather than only integer coefficients, so that the value $P(y)$ is still an integer for the $y$ the theorem produces.

Editorial extensions

If this is right

For every finite coloring, the monochromatic configuration $A \cup (A+B) \cup (A \cdot B)$ exists with $A$ infinite and $B$ of arbitrary finite size; whether $B$ can also be chosen infinite remains the open infinite version of the problem.
If the classical Pythagorean equation $x^2+y^2=z^2$ is ever shown to be $r$-regular for some $r>2$, then both perturbed equations of Theorem 1.4 are $r$-regular as well, because the proof transfers any such regularity bound directly.
The degree-of-regularity spectrum for nonlinear homogeneous equations is unbounded in every specified degree: for any $m$, taking $n=1,2,3,\ldots$ yields $m$-degree equations with arbitrarily large prescribed degree of regularity.
The $m=1$ case of Theorem 1.5 gives a route to the known linear degree-of-regularity result, and the $p$-adic valuation coloring provides a uniform obstruction for the non-regularity side of the question.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial: the proof's reliance on rational polynomial van der Waerden suggests a concrete repair: prove the theorem for rational-coefficient polynomials that take integer values on the relevant progression; until then, Theorem 1.2's proof is conditional on that extension.
Editorial: the formal Theorem 1.5 requires $\gcd(m,n)=1$, so the abstract's 'for every $m,n$' formulation needs an additional argument for pairs where $\gcd(m,n+1)>1$; a reader checking the reindexing will want that step supplied.
Editorial: the $p$-adic coloring used for non-regularity may extend to other prime-based coefficient families, giving exact degrees of regularity for a broader class of nonlinear homogeneous equations than the one constructed here.
Editorial: Theorem 1.4's transfer statement implies that progress on the classical Pythagorean triple question would automatically improve the perturbed equations, coupling the two problems in one direction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper announces two new results (Theorems 1.4 and 1.5) and a short proof of Bowen's theorem, but the central technical engine — a rational polynomial van der Waerden theorem — is false as stated. The proofs of the main theorems are therefore not valid in the present form.

What's genuinely good: the statements are interesting and the non-regularity direction of Theorem 1.5 (the p-adic valuation argument) is clean and correct. The idea of using homogeneous families to transfer polynomial van der Waerden to nonlinear equations is a sensible approach. The paper also correctly identifies that Theorem 1.2 is already due to Bowen and positions its contribution as a new proof.

The soft spots are serious and load-bearing. Theorem 2.1 claims a polynomial van der Waerden theorem for arbitrary rational polynomials with zero constant term. The proof applies Theorem 2.2 (the integer-polynomial theorem) to sets of the form A − P(n). When P(n) is not an integer, A − P(n) is not a subset of the positive integers, so the expression is undefined. Theorem 3.2 repeats the same error; the stress-test counterexample is decisive: take F={x/2} and color by parity. The pattern is never monochromatic. These failures propagate: the proofs of Theorems 1.2, 1.4(1), 1.4(2), and the n−1-regularity half of Theorem 1.5 all invoke either Theorem 2.1 or Theorem 3.3 directly. Lemma 3.4 also omits a divisibility condition on the polynomial family. The statement of Theorem 1.5 is internally inconsistent: it says 'n−1 regular but not n regular' then immediately says 'n regular but not n−1 regular'; the abstract says (n+1). That's trivial to fix but needs doing.

None of these flaws are beyond repair. The likely fix is to restrict all polynomial van der Waerden applications to integer-valued polynomials (or to integer-valued multiples after scaling), and to add explicit divisibility hypotheses to Lemma 3.4. The non-regularity part of Theorem 1.5 stands independently. I'd encourage a serious referee to work through the new statements with an eye toward those repairs; the paper is premature but not trivial.

Bottom line: if the authors fix the rational-polynomial gap and the statement inconsistency, Theorems 1.4 and 1.5 would be meaningful contributions. As it stands, the results are unproven. I would accept it for peer review rather than desk-reject, because the ideas are there and a careful referee might help the authors find the right framework.

Referee Report

6 major / 4 minor

Summary. The paper develops rational-polynomial analogues of Moreira's theorem and of the polynomial van der Waerden theorem, then applies them to three problems in arithmetic Ramsey theory. Theorem 1.2 claims that in every finite coloring of the positive integers one can find an infinite set A and an arbitrarily large finite set B such that A ∪ (A+B) ∪ A·B is monochromatic. Theorem 1.4 claims 2-regularity of equations of the form X^2+Y^2 = Z^2 + c·U^n V and X^2+Y^2 = Z^2 + c·(U±V). Theorem 1.5 claims, for every prime p and gcd(m,n)=1, a nonlinear homogeneous equation that is (n−1)-regular but not n-regular, yielding a nonlinear analogue of Rado's conjecture.

Significance. If the results were valid, they would be significant: Theorem 1.2 addresses a question of Kra, Moreira, Richter, and Robertson, and Theorem 1.5 would give a nonlinear counterpart to Alexeev–Tsimerman and Golowich. The paper also has positive features: it attempts to combine ultrafilter methods, piecewise syndetic sets, and p-adic valuations, and the p-adic nonregularity arguments in Section 4 appear internally coherent. However, the central rational-polynomial theorems are false as stated and are used at load-bearing points in the proofs of all three advertised applications, so the manuscript as written does not establish its main claims.

major comments (6)

[Section 2, Theorem 2.1] Theorem 2.1 is false as stated. Take F={x/2} and the parity coloring. If y is odd, then x+y/2 is not an integer, so the configuration is not defined in Z+; if y is even, say y=2y', then x and x+y' have opposite parity, so the three elements x, xy, x+y/2 are never monochromatic. The proof applies Theorem 2.2 to arbitrary rational polynomials, but the cited theorem is normally stated for integer-valued polynomials, and the proof never establishes that P(y) is an integer for the produced y. This directly invalidates the proof of Theorem 1.2, whose family F contains the rational linear polynomials (m/n)z for m,n in [1,R].
[Section 3, Theorem 3.2] Theorem 3.2 is false as stated for the same reason: for F={x/2} and the parity coloring, the pattern {d,a,a+d/2} is either undefined when d is odd or has a and a+d/2 of opposite parity when d is even. The failure is not repaired by adding a divisibility condition on d, because d even still forces the parity obstruction. Thus this theorem is not a harmless 'rational polynomial' generalization of the polynomial van der Waerden theorem and cannot be used as a black box for rational families.
[Section 3, Theorem 3.3; Section 4, proof of Theorem 1.5] The proof of Theorem 3.3 constructs a family F1 that contains rational linear polynomials such as (1/y)P(z/q), even when the original family F has integer coefficients. Applying the false Theorem 3.2 to F1 is therefore illegitimate. Since the claimed (n−1)-regularity in Theorem 1.5 is obtained by applying Theorem 3.3, the main degree-of-regularity result is not established. In addition, the sentence 'Since d1∈P, we can choose d1 such that q|d1' is not justified; divisibility by q must be imposed in the application of Theorem 3.2, not inferred from the conclusion.
[Section 3.2, Lemma 3.4 and Theorem 1.4(2)] Lemma 3.4 is unsupported because its proof applies Theorem 2.1 to the rational family F1={(1/n)P : P∈F, n∈[1,R]}, and Theorem 2.1 is false. The proof of Theorem 1.4(2) relies on Lemma 3.4, so the claimed 2-regularity of X^2+Y^2 = Z^2 + c·(U±V) is not established. The lemma's statement also uses the phrase 'is partition regular' where the proof appears to require monochromaticity; the intended meaning should be clarified.
[Section 3.1, proof of Theorem 1.4(1)] The proof of Theorem 1.4(1) is incomplete. It does not specify which theorem produces the monochromatic pattern {u, x+c u^n/2, x+c u^n/4, y,z}; the homogeneous family S is not defined at that point in the proof; and the displayed polynomials are rational-valued, so they fall into the false rational-polynomial regime. Even granting the existence of a monochromatic Pythagorean triple, the passage from that triple to a monochromatic solution of X^2+Y^2=Z^2+c U^n V is not rigorously justified as written.
[Section 4, Corollary 4.1 and Theorem 1.5] The p-adic arguments used to prove nonregularity appear internally consistent, but they only establish nonregularity. The matching lower bounds ('(n−1)-regular') depend on Theorem 3.3 and hence on the false rational-polynomial theorems above. Consequently the claimed conclusions of Corollary 4.1 and Theorem 1.5 are not established by the manuscript.

minor comments (4)

[Throughout] The symbol P is used both for the class of rational polynomials and for the set P defined in the proof of Theorem 1.2; these should be distinguished to avoid confusion.
[Theorem 1.5 and abstract] There is an indexing inconsistency: Theorem 1.5 states 'n−1-regular but not n-regular', while the abstract states 'n-regular but not (n+1)-regular', and the introductory sentence after Theorem 1.5 reverses the claim as 'n-regular but not n−1-regular'. The intended indexing should be stated uniformly.
[Proof of Theorem 1.2] In the membership argument for A+(B), the expression a(x+(d^i/a)y) is asserted to lie in D because a,d^i≤R, but this also requires (d^i/a)y to be an integer and the corresponding element to belong to C or P; neither condition is established.
[Abstract and introduction] The paper calls Theorem 1.2 a 'finitary version' of the Kra–Moreira–Richter–Robertson question, but A is still required to be infinite; the terminology should be clarified or adjusted.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular dependence found; derivation chain rests on external theorems and internally proved lemmas.

full rationale

The central claims (Theorems 1.2, 1.4, and 1.5) are derived from external results: the polynomial van der Waerden theorem (Theorem 2.2, cited to Hindman [24]), ultrafilter facts ([25,26]), the 2-regularity of Pythagorean triples ([20]), and elementary p-adic valuation arguments. Theorem 2.1 is proved by adapting Moreira's proof [29], not by assuming the target. Theorem 3.3 is proved from Theorem 3.2, whose proof uses the polynomial van der Waerden theorem; the applications then invoke Theorem 3.3, so the chain is linear and internally supported. I found no fitted parameter renamed as prediction, no definition that assumes the target, no load-bearing self-citation (no self-citations by the authors), and no uniqueness theorem imported from the authors' prior work. The mathematical weaknesses noted by reviewers—whether polynomial van der Waerden applies to arbitrary rational polynomials, whether the compactness step in Theorem 3.3 is valid, and whether Theorem 3.2 fails for F={x/2}—are correctness concerns rather than circularity concerns: they question whether the hypotheses used are true, not whether the conclusions are assumed as inputs. Hence no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's central claims depend on polynomial van der Waerden theorems, ultrafilter theory, and the 2-regularity of Pythagorean triples. The most fragile input is the unrestricted rational-polynomial version of the van der Waerden theorem, which appears false or at least unproved as stated.

assumptions (4)

ad hoc to paper The polynomial van der Waerden theorem holds for finite families of rational polynomials with zero constant term, and the chosen parameter n makes each P(n) an integer.
Theorem 2.1 and Lemma 3.4 apply Theorem 2.2 to rational polynomials such as (m/n)x. Standard forms of the polynomial van der Waerden theorem require integer-valued polynomials; no integrality of P(n) is established in the paper.
domain assumption The homogeneous polynomial van der Waerden theorem (Theorem 3.3) can be applied with d divisible by an arbitrary m, which would clear denominators in F.
The theorem's 'in addition' clause provides m|d, but the proofs do not consistently invoke it. Lemma 3.4 omits this clause entirely, making it false as stated.
domain assumption Pythagorean triples form a homogeneous 2-regular family, as established by Heule, Kullmann, and Marek.
Used in Section 3 to define the homogeneous family S for the proof of Theorem 1.4.
standard math Basic p-adic valuation properties, including multiplicativity and the additive property for terms with distinct valuations.
Used in the non-regularity part of Theorem 1.5 and Corollary 4.1 to derive contradictions from distinct valuations.

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Pith. "Pith review of Homogeneous Patterns in Ramsey Theory." pith.science (2026). https://pith.science/paper/AKZF6BBG

@misc{pith2026250117203,
  author       = {Pith},
  title        = {Pith review of: Homogeneous Patterns in Ramsey Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AKZF6BBG}},
  note         = {Machine review of arXiv:2501.17203}
}

abstract

In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for every finite coloring of $\mathbb{Z}^+$, there exist an infinite set $A$ and an arbitrarily large finite set $B$ such that $A \cup (A+B) \cup A \cdot B$ is monochromatic. This result resolves the finitary version of a question posed by Kra, Moreira, Richter, and Robertson regarding the partition regularity of $(A+B) \cup A \cdot B$ for infinite sets $A, B$ (see (Question 8.4, J. Amer. Math. Soc., 37 (2024))), which is closely related to a question of Erd\H{o}s. As the second application, we make progress on a nonlinear extension of the partition regularity of Pythagorean triples. Specifically, we demonstrate that the equation $x^2 + y^2 = z^2 + P(u_1, \dots, u_n)$ is $2$-regular for certain appropriately chosen polynomials $P$ of any desired degree. Finally, as the third application, we establish a nonlinear variant of Rado's conjecture concerning the degree of regularity. We prove that for every $m, n \in \mathbb{Z}^+$, there exists an $m$-degree homogeneous equation that is $n$-regular but not $(n+1)$-regular. The case $m = 1$ corresponds to Rado's conjecture, originally proven by Alexeev and Tsimerman (J. Combin. Theory Ser. A, 117 (2010), and later independently by Golowich (Electron. J. Combin. 21 (2014)).

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Alweiss: Monochromatic sums and products of polynomials, Discrete Anal

R. Alweiss: Monochromatic sums and products of polynomials, Discrete Anal. 2024:5, 7 pp

work page 2024

[2] [2]

Alexeev and J

B. Alexeev and J. Tsimerman: Equations resolving a conjecture o f Rado on partition regularity, J. Combin. Theory Ser. A 117 (2010), 1008–1010

work page 2010

[3] [3]

Bergelson: Ergodic Ramsey theory—an update, in Ergodic Theory of Zd Actions (Warwick, 1993–1994), London Math

V. Bergelson: Ergodic Ramsey theory—an update, in Ergodic Theory of Zd Actions (Warwick, 1993–1994), London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 1–61

work page 1993

[4] [4]

Bergelson and A

V. Bergelson and A. Leibman: Polynomial extensions of van der Wa erden and Szemer´ edi theorems, J. Amer. Math. Soc. 9 (1996), 725–753

work page 1996

[5] [5]

Bergelson, J

V. Bergelson, J. H. Johnson Jr., and J. Moreira: New polynomial a nd multidimensional extensions of classical partition results, J. Combin. Theory Ser. A 147 (2017), 119–154

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Bowen: Monochromatic products and sums in 2-colorings of N, Adv

M. Bowen: Monochromatic products and sums in 2-colorings of N, Adv. Math. 462 (2025), 110095

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Bowen: Composite Ramsey theorems via trees, arXiv:2210.14311v2

M. Bowen: Composite Ramsey theorems via trees, arXiv:2210.14311v2

arXiv

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Erd˝ os: Problems and results on combinatorial number theor y

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