{"id":"cb3b3fee-e784-4457-90a8-30c5667a92e6","arxiv_id":"2501.17203","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper establishes new partition regularity results for nonlinear equations, including m-degree homogeneous equations with prescribed degree of regularity.","lead":"This paper proves three Ramsey-theoretic results, including a nonlinear version of Rado's conjecture on the degree of regularity of homogeneous equations. The proofs use polynomial van der Waerden theorems and ultrafilter methods, but several statements about rational polynomials are not justified as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2 is false for rational polynomials: F={x/2} fails under parity coloring, and the proof of Theorem 1.5 relies on this lemma via Theorem 3.3.","rationale":"The reader's weakest assumption concerned Theorem 2.1's use of rational polynomials in the polynomial van der Waerden theorem. The most load-bearing problem for the central claim is a concrete instance of the same disease: Theorem 3.2, which is the lemma actually used in the proof of Theorem 1.5 through Theorem 3.3, is false as stated. The counterexample is simple and decisive. This does not necessarily show Theorem 1.5 is false; the construction might be repairable by restricting to integer-valued polynomials or by clearing denominators before applying polynomial van der Waerden, but it means the proof in the manuscript is invalid. The reader's verdict of CONDITIONAL remains appropriate: the paper needs a corrected lemma or a different proof of the homogeneous polynomial van der Waerden step. Secondary issue: the abstract's 'for every m,n' does not follow from Theorem 1.5 because replacing n by n+1 requires gcd(m,n+1)=1; this is another reason the advertised theorem is not established, but it is less immediate than the false lemma.","tokens_in":11277,"tokens_out":22406,"duration_ms":197729,"concrete_test":"Verify the counterexample: for F={x/2} under the 2-coloring by parity, enumerate all d,a in {1,...,N} and confirm the set {d,a,a+d/2} is never monochromatic (when defined). Analytically, d odd gives a+d/2 not in Z, and d=2d' gives a parity conflict. Then check whether the proof of Theorem 3.3 can be modified to avoid rational polynomials, or whether Theorem 1.5's regularity argument can be rerun with an integer-valued polynomial family; if not, the main theorem is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.2 claims that for any finite set F of rational polynomials with zero constant term and any finite coloring of Z+, there are a,d in Z+ such that {d} union {a+P(d): P in F} is monochromatic. This is false. Take F={x/2} and the parity coloring. The pattern is {d,a,a+d/2}. If d is odd, a+d/2 is not an integer, so the configuration is undefined in Z+. If d=2d', then 2d' is even while a and a+d' have opposite parity; hence the three elements are never all the same color. Thus Theorem 3.2 fails. The proof of Theorem 1.5 invokes Theorem 3.3, and the proof of Theorem 3.3 applies Theorem 3.2 to a family F1 that necessarily contains rational linear polynomials such as (1/b_i)P(b_1 x/q). Therefore the central proof of Theorem 1.5's n-1-regularity rests on a false statement and is not valid as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11517,"tokens_out":12198,"duration_ms":121931,"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":[{"comment":"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":"Section 2, Theorem 2.1"},{"comment":"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":"Section 3, Theorem 3.2"},{"comment":"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":"Section 3, Theorem 3.3; Section 4, proof of Theorem 1.5"},{"comment":"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":"Section 3.2, Lemma 3.4 and Theorem 1.4(2)"},{"comment":"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":"Section 3.1, proof of Theorem 1.4(1)"},{"comment":"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.","section":"Section 4, Corollary 4.1 and Theorem 1.5"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Theorem 1.5 and abstract"},{"comment":"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.","section":"Proof of Theorem 1.2"},{"comment":"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.","section":"Abstract and introduction"}],"recommendation":"reject","confidential_remarks":"The central rational-polynomial lemmas (Theorems 2.1 and 3.2) are false as stated, and they are used repeatedly in the proofs of the paper's advertised applications. This is not a local, fixable gap: the claimed theorems fail on elementary counterexamples, and the proof strategy would require substantial reworking rather than small corrections. I recommend rejection, though a substantially revised version that restricts to integer-valued polynomials or proves a correct rational variant might contain viable ideas."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nThe 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.\n\nNone 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.\n\nBottom 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.","headline":"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.","tokens_in":12052,"tokens_out":4266,"would_cite":false,"duration_ms":38168,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","05C55","22A15","54D35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonlinear equations can be built with any prescribed degree of regularity.","keywords":["Ramsey theory","partition regularity","homogeneous patterns","polynomial van der Waerden theorem","Pythagorean triples","degree of regularity","p-adic valuations"],"falsifier":"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.","tokens_in":11073,"feed_emoji":"🎨","tokens_out":14659,"duration_ms":134852,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every finite coloring hides monochromatic A+B and A·B","feed_subtitle":"New homogeneous-pattern proof also tunes nonlinear equations to any chosen degree of regularity.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the polynomial van der Waerden theorem (Theorem 3.6) that the paper extends to rational polynomials and uses throughout.","marker":"[24]"},{"why":"Provides the proof strategy and monochromatic sums-and-products result that Theorem 2.1 adapts.","marker":"[29]"},{"why":"Poses the open question on infinite sets $A$ and $B$ with $A+B$ and $A\\cdot B$ monochromatic, whose finitary version is resolved.","marker":"[27]"},{"why":"Introduces homogeneous sets for degree-of-regularity arguments and proves the linear counterpart the paper generalizes.","marker":"[13]"},{"why":"Gives the original resolution of the linear degree-of-regularity conjecture that serves as the $m=1$ baseline.","marker":"[2]"},{"why":"Establishes the Boolean Pythagorean triples result that supplies the homogeneous 2-regular family used in Theorem 1.4.","marker":"[20]"},{"why":"Provides the Stone-Cech compactification framework, piecewise syndetic sets, and compactness arguments used in Sections 2 and 3.","marker":"[25]"},{"why":"Supplies Lemma 1.5 used to keep scaled sets piecewise syndetic in the proof of Theorem 2.1.","marker":"[26]"},{"why":"Formulates the original degree-of-regularity conjecture and the linear partition-regularity background the paper extends.","marker":"[30]"}],"fun_headline_variants":["Any finite coloring hides monochromatic A, A+B, and A·B","Nonlinear Rado's conjecture: every degree and regularity achieved","Homogeneous Ramsey: infinite monochromatic patterns from finite colorings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Any finite coloring hides monochromatic A, A+B, and A·B","Nonlinear Rado's conjecture: every degree and regularity achieved","Homogeneous Ramsey: infinite monochromatic patterns from finite colorings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001467,"raw_usage":{"total_tokens":5977,"prompt_tokens":1099,"completion_tokens":4878,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":715,"completion_tokens_details":{"reasoning_tokens":4819}},"tokens_in":715,"tokens_out":4878,"duration_ms":40859,"temperature":1.0,"reasoning_tokens":4819,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T12:06:16.697955+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hindman: Problems and new results in the algebra of Beta S and Ramsey theory, in Unsolved Problems in Mathematics for the 21st Century , J","cited_arxiv_id":null,"evidence_quote":"Supplies the polynomial van der Waerden theorem (Theorem 3.6) that the paper extends to rational polynomials and uses throughout."},{"cited_title":"Moreira: Monochromatic sums and products in N, Ann","cited_arxiv_id":null,"evidence_quote":"Provides the proof strategy and monochromatic sums-and-products result that Theorem 2.1 adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Poses the open question on infinite sets $A$ and $B$ with $A+B$ and $A\\cdot B$ monochromatic, whose finitary version is resolved."},{"cited_title":"Golowich: Resolving a conjecture on degree of regularity of lin ear homogeneous equations, Electron","cited_arxiv_id":null,"evidence_quote":"Introduces homogeneous sets for degree-of-regularity arguments and proves the linear counterpart the paper generalizes."},{"cited_title":"Alexeev and J","cited_arxiv_id":null,"evidence_quote":"Gives the original resolution of the linear degree-of-regularity conjecture that serves as the $m=1$ baseline."},{"cited_title":"Heule, O","cited_arxiv_id":null,"evidence_quote":"Establishes the Boolean Pythagorean triples result that supplies the homogeneous 2-regular family used in Theorem 1.4."},{"cited_title":"Hindman and D","cited_arxiv_id":null,"evidence_quote":"Provides the Stone-Cech compactification framework, piecewise syndetic sets, and compactness arguments used in Sections 2 and 3."},{"cited_title":"Hindman and D","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 1.5 used to keep scaled sets piecewise syndetic in the proof of Theorem 2.1."}],"review_version":1}