Matrix Formulation of Moreira Theorem

Sayan Goswami

REVIEW 2 major objections 3 minor 18 references

Matrix Formulation of Moreira Theorem

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that any two finite image partition regular matrices of the same order force a monochromatic triple {AX, AX+BY, AX·BY}.

desk verdict A natural matrix version of Moreira's theorem, but the proof has a fatal divisibility gap that breaks the main reduction. read the letter →

arxiv 2501.16595 v1 pith:MZFOBZWO submitted 2025-01-28 math.CO

Sayan Goswami This is my paper · ORCID

classification math.CO MSC 05D10

keywords partitionregularityimageregularmatricesMoreira'stheoremmonochromaticsumsandproductsStone–Čechcompactificationultrafiltersfinitecoloringsofnaturals

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 sets out to prove a matrix version of Moreira's theorem. It claims that if $A$ and $B$ are two finite image partition regular matrices—matrices whose images always contain an all-one-color vector in every finite coloring—with the same number of rows, then every finite coloring of $\mathbb{N}$ has vectors $X,Y$ for which the three vectors $AX$, $AX+BY$, and $AX\cdot BY$ are monochromatic under coordinatewise addition and multiplication. In fact it claims the stronger scalar statement that the set of all numbers $a$, $a+b$, and $ab$ with $a$ an entry of $AX$ and $b$ an entry of $BY$ lies in one color. The scalar case $A=B=(1)$ recovers Moreira's theorem that $\{x,x+y,xy\}$ is monochromatic, so the result extends a known sum-product pattern to a whole class of arithmetic configurations.

What carries the argument

Three pieces carry the argument. A finite image partition regular matrix is one whose image, for every finite coloring, contains an all-one-color vector; the theorem ranges over such matrices with a common row count. Lemma 2.1, proved with minimal idempotent ultrafilters in the Stone–Čech compactification $\beta\mathbb{N}$, puts $\{AX, BY, AX\cdot BY\}$ into one color class. Theorem 1.2, a polynomial form of Moreira's theorem, supplies infinitely many $x$ such that $\{x, xy, x+f(y): f\in F\}$ is monochromatic; the proof builds an auxiliary family $F'$ of rational dilations of polynomials so that coefficients like $b/a$ can be realized inside that family. A compactness reduction turns the infinite coloring into a finite interval $[1,R]$, allowing the monochromatic triple from Lemma 2.1 to be scaled into the same color class.

What would settle it

A concrete test is to take a small two-row finite image partition regular matrix such as $A=B=\begin{pmatrix}1&0\\1&1\end{pmatrix}$, choose a finite coloring, run the construction, and check whether the auxiliary $y$ from the polynomial Moreira theorem makes $x+\frac{b}{a}y$ an integer for every $a\in AX$ and $b\in BY$; an instance where it is not would break the containment $AX+BY\subseteq D$ and show that the present proof needs a new argument.

Watch

Extended reading notes

Core claim

Moreira's theorem says that in every finite coloring of the naturals there are $x,y$ with $\{x,x+y,xy\}$ monochromatic. The central claim here is the matrix generalization: for any finite image partition regular matrices $A$ and $B$ with the same number of rows, there are vectors $X$ and $Y$ such that $\{AX, AX+BY, AX\cdot BY\}$ is monochromatic, with addition and multiplication performed coordinatewise. The author states a stronger form: the full set $\{a, a+b, ab : a \text{ an entry of } AX,\ b \text{ an entry of } BY\}$ is monochromatic. Taking $A=B=(1)$ gives exactly Moreira's configuration, so the theorem contains the scalar result as a special case.

Load-bearing premise

The proof's final containment step assumes that for every entry $a$ of $AX$ and $b$ of $BY$, the quantity $x+\frac{b}{a}y$ is a natural number; nothing in the argument shows that $a$ divides $by$.

Editorial extensions

If this is right

For $A=B=(1)$, the theorem reduces to Moreira's theorem: $\{x,x+y,xy\}$ is monochromatic in every finite coloring of $\mathbb{N}$.
For the matrices whose images are arithmetic progressions, the result gives a monochromatic configuration containing an arithmetic progression, a second arithmetic progression shifted by the first, and all coordinatewise products of their entries.
The stronger scalar conclusion means the monochromatic pattern includes every number of the form $a$, $a+b$, or $ab$ formed from the entries of the two matrix images, not just the three whole vectors.
Because the theorem applies to any two finite image partition regular matrices, it places several previously separate Ramsey-theoretic patterns under one matrix statement.

Reading between the lines

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

The author leaves the infinite-matrix case open; extending the argument to the infinite finite-sums matrix would connect the theorem to the finite-sums formulation of Ramsey theory.
Choosing $A$ and $B$ to be distinct arithmetic-progression matrices suggests new concrete monochromatic configurations that have not been isolated explicitly in the scalar literature.
A natural test of the construction is whether the auxiliary $y$ from the polynomial Moreira theorem can always be chosen divisible by each entry of $AX$; if it can, the proof becomes fully constructive for those cases.

Share X Bluesky LinkedIn Reddit HN

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

The one thing you should know: this paper states a genuinely new matrix generalization of Moreira's theorem, but the proof collapses at a single divisibility step, so the main theorem is not established as written.

What is actually new: the statement itself. Nobody in the cited literature formulates Moreira's theorem for finite image partition regular matrices, and the scalar case A=B=(1) recovers {x, x+y, xy}. The compactness argument and the ultrafilter lemma (Lemma 2.1) are sketched briskly but look plausible; Lemma 2.1 is a standard sort of central-set argument, and the claimed stronger set-wise formulation is a reasonable extension. The paper is short, clear, and honestly acknowledges the infinite-matrix question at the end.

The soft spot is fatal. In the proof of Theorem 1.3, the author wants to show that AX + BY lies inside the monochromatic set D. For each a in AX and b in BY, he writes a(x + (b/a)y) = ax + by, and asserts that x + (b/a)y is in the monochromatic polynomial set P. That requires (b/a)y to be a natural number, i.e., y divisible by a. No divisibility guarantee is supplied. The stress-test example is exactly right: with A=(2), B=(1), you need y even, and nothing in the construction of y forces that. The auxiliary family F' also uses rational-coefficient polynomials P(y/z · x), which need not take natural values, so even applying Theorem 1.2 is questionable. These two problems compound; the first alone is enough to break the containment AX + BY ⊆ D. The compactness step is terse but likely repairable; the divisibility issue is not a minor fix, it is the hinge of the whole reduction.

Who this is for: arithmetic Ramsey theorists who care about partition regularity of {x,y,x+y,xy} and matrix generalizations. The formulation is a good thing to have in the literature as a conjecture or a program, but the proof does not support the theorem. If you need the result, start from Lemma 2.1 and look for another route to the additive-multiplicative pattern.

Recommendation: send it to a serious referee, but expect the verdict to be reject unless the divisibility gap is repaired. The idea is worth refereeing; the current proof is not.

Referee Report

2 major / 3 minor

Summary. The paper claims a matrix generalization of Moreira's theorem. Specifically, it asserts (Theorem 1.3) that if A and B are finite image partition regular matrices of the same order, then every finite coloring of N contains a monochromatic set of the form {AX, AX+BY, AX·BY}, with coordinatewise operations, and a stronger version in which all elements {a, a+b, ab} with a ∈ AX and b ∈ BY are monochromatic. The proof strategy is to first prove Lemma 2.1, which gives a monochromatic configuration {AX, BY, AX·BY} using a minimal idempotent ultrafilter, and then to combine this with a polynomial version of Moreira's theorem (Theorem 1.2) to force AX+BY into the same color class.

Significance. If the main theorem were proved, it would be a natural and substantial extension of Moreira's theorem and would unify several matrix-based Ramsey-theoretic results. The paper is concise, and Lemma 2.1 is a plausible and standard ultrafilter argument that is likely correct. The claimed result is also stated honestly, including the limitation that infinite matrices are not treated. However, the proof of Theorem 1.3 has a load-bearing gap: the transition from the polynomial family to the containment AX+BY ⊆ D requires a divisibility condition that is never established and is in fact not guaranteed by the preceding construction. As a result, the central theorem is not proved in this manuscript.

major comments (2)

[Section 2, proof of Theorem 1.3] The containment A(xX)+B(yY) ⊆ D is asserted from the identity a(x+(b/a)y)=ax+by. For this to place ax+by in D = {AX, BY, AX·BY}·P, the element x+(b/a)y must belong to the monochromatic set P = {x, xy, x+P(y) : P ∈ F}. This requires (b/a)y to be one of 0, y, or P(y) for some P ∈ F, and in particular requires (b/a)y to be a natural number. No divisibility of y by a is established. The entries a and b are produced later by the finite coloring argument, after y has been fixed by Theorem 1.2, so y cannot be chosen to satisfy these divisibility conditions. For example, with A=(2) and B=(1), the argument would need y to be even, but no such condition is supplied. Without this, x+(b/a)y may not even be a natural number, so the set membership and the coloring argument are not meaningful.
[Section 2, definition of F' and application of Theorem 1.2] The family to which Theorem 1.2 is applied is F' = { (1/y) P_{z/q} : P ∈ F, y,z ∈ [1,R] }. This is not an admissible input for Theorem 1.2 as stated. First, the symbol F is not defined in the statement or proof of Theorem 1.3, so the finite family is ambiguous; if F is intended to be the set of all polynomials, then F' is not finite as required. Second, and more seriously, the polynomials in F' generally have rational coefficients, so their values at natural arguments need not be natural numbers. Theorem 1.2, and the quoted result [10, Proof of Theorem 1.4], concern polynomials whose values at the relevant variable are natural, since expressions such as x+P(y) must be elements of the colored set N. The paper neither proves a rational-coefficient variant nor explains why the rational coefficients are harmless. This gap is load-bearing because F' is the only mechanism intended to produce elements of the form x+(b/a)y in P.

minor comments (3)

[Section 2, proof of Theorem 1.3, first paragraph] The compactness step asserting the existence of R with the stated finite property needs a justification: one must also bound the entries of AX, BY, and AX·BY by R. This is a standard compactness argument, but it is not stated.
[Section 2, definition of the coloring χ] The definition χ(m)=ω(m·P) uses P as a set, not a number. The intended meaning is that χ(m)=ω(m·p) for any (equivalently every) p ∈ P, using the fact that P is monochromatic under ω1; this should be written explicitly.
[Throughout] There are several typographical errors and garbled formulas, including 'col o ring', 'W est Benagal', the notation 'a, d P r1, R s' for the vector variables, and the formula for F'. These should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof imports Moreira's theorem and standard ultrafilter facts as black boxes, and its central reduction does not define the target pattern in terms of its own conclusion.

full rationale

The paper's derivation chain is not circular. Theorem 1.3 is proved assuming Lemma 2.1, and Lemma 2.1 is proved using standard ultrafilter facts from Hindman–Strauss (an external, published reference) together with the definition of image partition regular matrices. The only self-citation is the introductory remark that the authors previously proved a set-theoretic version of Moreira's theorem, but this is not used as a load-bearing ingredient in the proof of Theorem 1.3 or Lemma 2.1. No parameter is fitted to the target conclusion, and the target set {AX, AX+BY, AX·BY} is not defined in terms of the sets whose monochromaticity is being proved. The proof does contain a likely correctness gap: the containment AX + BY ⊆ D requires x + (b/a)y to lie in the monochromatic set P for each a ∈ AX and b ∈ BY, which needs a divisibility condition that is not established. However, that is a mathematical flaw, not a circularity: the proof does not assume Theorem 1.3 to prove itself, and the gap does not amount to renaming a fitted input as a prediction. The reliance on Moreira's theorem as an external black box is legitimate support rather than circular reasoning.

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

No free parameters or invented entities appear. The proof leans on Moreira's theorem and ultrafilter machinery from [9]. Two additional assumptions are needed but not justified: integer-valuedness and applicability of the rational polynomial family, and divisibility of the Moreira y by all entries of A X.

assumptions (4)

standard math Theorem 1.2 of Moreira: for any finite family F of polynomials with no constant term, some y has infinitely many x with {x, xy, x+f(y)} monochromatic.
External theorem from [10] used as a black box. It is not circular, but it is load-bearing for the proof of Theorem 1.3.
standard math Standard ultrafilter facts from [9], including the characterization of finite image partition regular matrices by minimal idempotent ultrafilters.
Used in Lemma 2.1 to build the vectors X and Y from a minimal idempotent ultrafilter.
ad hoc to paper The auxiliary family F' is contained in the set of polynomials P to which Theorem 1.2 applies.
This is asserted but not established. The expressions in F' involve rational coefficients and divisions, and integer-valuedness on N is not shown.
ad hoc to paper The y from Moreira's theorem in the proof is divisible by every entry a of A X, so that (b/a)y is an integer.
Needed in the step a(x+(b/a)y)=ax+by. No proof or cited theorem gives this divisibility, and it is not a consequence of the statements in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Matrix Formulation of Moreira Theorem." pith.science (2026). https://pith.science/paper/MZFOBZWO

@misc{pith2026250116595,
  author       = {Pith},
  title        = {Pith review of: Matrix Formulation of Moreira Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZFOBZWO}},
  note         = {Machine review of arXiv:2501.16595}
}

abstract

In a celebrated article, Moreira proved for every finite coloring of the set of naturals, there exists a monochromatic copy of the form $\{x,x+y,xy\},$ which gives a partial answer to one of the central open problems of Ramsey theory asking whether $\{x,y,x+y,xy\}$ is partition regular. In this article, we prove the matrix version of the Moreira theorem. We prove that if $A$ and $B$ are two finite image partition regular matrices of the same order, then for every finite coloring of the set of naturals, there exist two vectors $\overrightarrow{X}, \overrightarrow{Y}$ such that $\{A\overrightarrow{X}, A\overrightarrow{X}+B\overrightarrow{Y}, A \overrightarrow{X}\cdot B\overrightarrow{Y}\}$ is monochromatic, where addition and multiplication are defined coordinate-wise.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 16 canonical work pages

[1]

S. D. Adhikari, and S. Goswami: Homogeneous patterns in Ramsey theory, ArXiV: submit/6159148

work page arXiv
[2]

Alweiss: Monochromatic sums and products of polynomials, Discrete Anal

R. Alweiss: Monochromatic sums and products of polynomials, Discrete Anal. 2024:5, 7 pp. 2

work page 2024
[3]

Alweiss: Monochromatic sums and products over Q

R. Alweiss: Monochromatic sums and products over Q . Preprint (2023) arXiv:2307.08901

arXiv 2023
[4]

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

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

work page 1993
[5]

Bergelson, and J

V. Bergelson, and J. Moreira: Ergodic theorem involving additive and multiplicative groups of a field and patterns, Ergodic Theory Dynam. Systems, 37 (3), May 2017, pp. 673 - 692

work page 2017
[6]

Bergelson, and J

V. Bergelson, and J. Moreira: Measure preserving actions of affine semigroups and \ x+y,xy\ patterns, 38 (2), Ergodic Theory Dynam. Systems, April 2018 , 473 - 498

work page 2018
[7]

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

work page 2025
[8]

Composite Ramsey theorems via trees

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

work page Pith review arXiv

Show all 18 references

[9]

Erd o s: Problems and results on combinatorial number theory

P. Erd o s: Problems and results on combinatorial number theory. III, Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes in Math., Vol. 626, Springer, Berlin, 1977, pp. 43–72

1976
[10]

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
[11]

R. L. Graham, B. L. Rothschild, and J. H. Spencer: Ramsey Theory, second ed., Wiley-Intersci. Ser. Discrete Math. Optim., John Wiley & Sons, New York, 1990

1990
[12]

Hindman: Finite sums from sequences within

N. Hindman: Finite sums from sequences within
[13]

Hindman: Partitions and sums and products of integers, Trans

N. Hindman: Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227–245

1979
[14]

Hindman, I

N. Hindman, I. Leader, and D. Strauss: Open problems in partition regularity, Combin. Probab. Comput. 12 (2003), 571–583

2003
[15]

Hindman, and D

N. Hindman, and D. Strauss: Algebra in the Stone- C ech Compactifications: theory and applications, second edition, de Gruyter, Berlin, 2012

2012
[16]

Moreira: Monochromatic sums and products in N , Annals of Mathematics (2) 185 (2017), no

J. Moreira: Monochromatic sums and products in N , Annals of Mathematics (2) 185 (2017), no. 3, 1069–1090

2017
[17]

Rado: Studien zur Kombinatorik, Math

R. Rado: Studien zur Kombinatorik, Math. Z. 36 (1933), 242–280

1933
[18]

van der Waerden: Beweis einer baudetschen vermutung

B.L. van der Waerden: Beweis einer baudetschen vermutung. Nieuw. Arch. Wisk., 15 (1927), 212-216

1927

Pith tools

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

[1] [1]

S. D. Adhikari, and S. Goswami: Homogeneous patterns in Ramsey theory, ArXiV: submit/6159148

work page arXiv

[2] [2]

Alweiss: Monochromatic sums and products of polynomials, Discrete Anal

R. Alweiss: Monochromatic sums and products of polynomials, Discrete Anal. 2024:5, 7 pp. 2

work page 2024

[3] [3]

Alweiss: Monochromatic sums and products over Q

R. Alweiss: Monochromatic sums and products over Q . Preprint (2023) arXiv:2307.08901

arXiv 2023

[4] [4]

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

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

work page 1993

[5] [5]

Bergelson, and J

V. Bergelson, and J. Moreira: Ergodic theorem involving additive and multiplicative groups of a field and patterns, Ergodic Theory Dynam. Systems, 37 (3), May 2017, pp. 673 - 692

work page 2017

[6] [6]

Bergelson, and J

V. Bergelson, and J. Moreira: Measure preserving actions of affine semigroups and \ x+y,xy\ patterns, 38 (2), Ergodic Theory Dynam. Systems, April 2018 , 473 - 498

work page 2018

[7] [7]

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

work page 2025

[8] [8]

Composite Ramsey theorems via trees

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

work page Pith review arXiv