{"id":"0844cec4-c4e6-4c3e-aec4-f79bc58a48bf","arxiv_id":"2411.15987","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite colorings of G^2 for countable amenable G contain monochromatic right corners with IP* return times, and FC-groups satisfy the analogous statement in every dimension.","lead":"This paper proves a nonabelian Ramsey theorem: for any finite coloring of G^2 with G a discrete countable amenable group, one color class contains a monochromatic right corner, and the shifts producing such a corner form a left IP* set. It also proves the full-dimensional version for FC-groups, where every conjugacy class is finite.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A depends on an unproved projection lemma for arbitrary idempotent ultrafilters; the sketch in §2.2 is incomplete and contains an index-order slip.","rationale":"The reader's weakest assumption is correctly located: Section 2.2's Theorem 5 is the only place where the full IP* strength is obtained, and it depends on the arbitrary-idempotent projection lemma. I checked the surrounding argument: Theorem 6 transfers the IP* return set to positive-density sets in G^2, Theorem 13 supplies the hypothesis for the color-focusing lemma, Theorems 11-12 are internally consistent modulo the routine product-subsystem concatenation, and the FC-group extension via van der Waerden families is plausible. No other step appears comparably fragile. The projection lemma itself is likely true: the weak-operator compactification argument proves it in a few lines, so the concern is about completeness of the written proof rather than a demonstrated contradiction. Hence I partially agree with the reader: same location, but I do not see a falsity. The appropriate response is to keep the conditional verdict pending a full derivation of the projection lemma; once that derivation is added, the central proof appears sound.","tokens_in":17637,"tokens_out":46486,"duration_ms":421201,"concrete_test":"Independently derive the projection lemma without minimality: for any unitary representation U and any p in βG, define Φ(p) in the weak operator topology as p-lim U_g; prove Φ(p•q) = Φ(p)Φ(q) by checking the identity on principal ultrafilters and using continuity of q ↦ p•q. Then take p idempotent and verify Φ(p)^2 = Φ(p) and Φ(p)^* = Φ(p) via the standard idempotent-contraction argument. If the derivation succeeds, the §2.2 gap is closed and Theorem A stands; if the semigroup homomorphism identity fails for some representation or idempotent, isolate that counterexample and the IP* upgrade in Theorem 5 is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The upgrade from central* to IP* in Theorem A passes through Theorem 5, where P f = p-lim_g T_g f is required to be an orthogonal projection for every idempotent p in (βG, •), not only for minimal idempotents as in the cited Theorem 4. This is genuinely load-bearing: the displayed identity p-lim_g μ(A ∩ T_g^{-1}A) = ∫(P1_A)^2 dμ ≥ μ^2(A) > 0 uses P^2 = P and ||P|| ≤ 1 to conclude that the return set lies in p, hence in every idempotent. The paper's sketch for P^2 = P tries to show P^* = P^*P^* with p^{-1}, but the equality U_g^*U_h^* = U_{gh}^* is not justified for the unitary representation U_{gh} = U_gU_h defined in the proof; the correct identity is U_g^*U_h^* = U_{hg}^*, so idempotence is not established as written. The desired fact is nevertheless true: the map Φ(p) = WOT-p-lim U_g is a semigroup homomorphism from (βG, •) into the contractions of H, so p idempotent gives Φ(p)^2 = Φ(p), and a norm-bounded idempotent operator is self-adjoint. But this argument is absent from the paper. If the projection property were to fail for some idempotent p, the set {g : μ(A ∩ T_g^{-1}A) > 0} would not automatically belong to p, the IP* conclusion of Theorem 5 would collapse, and Theorem A would not follow from the presented argument.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove that for every countable, discrete, amenable group G, any finite coloring of G^2 contains a monochromatic right corner of the form {(x,y),(xg,y),(xg,yg)} for a set of shifts g that is left IP* (Theorem A). It further claims that for FC-groups, the analogous statement holds in G^m for every m (Theorem B), thereby upgrading a conjecture of Bergelson and McCutcheon from central* to IP* in these cases. The proof strategy is ergodic: an IP-Poincaré recurrence theorem for a single measure-preserving anti-action (Theorem 5) is combined with a Furstenberg correspondence principle and a color-focusing lemma (Theorem 11) to obtain the two-dimensional van der Waerden result; a van der Waerden family framework and piecewise syndetic sets are then used to handle FC-groups.","tokens_in":17981,"tokens_out":27492,"duration_ms":226615,"significance":"If the proofs are completed, Theorem A would be a genuine advance: it establishes the m=2 case of the Bergelson-McCutcheon conjecture for all countable amenable groups and strengthens the conclusion from central* to IP*. Theorem B extends the result to all dimensions for FC-groups, a natural class beyond abelian groups. The paper is conceptually clear and uses standard tools (Stone-Čech compactification, Furstenberg correspondence, Følner density), and the color-focusing framework is a useful contribution. However, the main novelty—passing from minimal idempotents to arbitrary idempotents in the recurrence theorem—is exactly where the proof is incomplete, so the significance is conditional on fixing that gap. No machine-checked proofs or code are supplied; the strength of the paper lies in its framework and the potential of its main theorems.","major_comments":[{"comment":"The paper's central upgrade from central* to IP* rests on the claim that for an arbitrary idempotent p in (βG,•), the operator P = p-lim_g T_g defined in Theorem 5 is an orthogonal projection. This claim is not proved. The discussion before Theorem 5 says the assumption of minimality in the cited [6, Theorem 2.4] is only needed for a commutation property and can be dropped, but no valid derivation is given. The displayed computation for P^*P^* is wrong: for a unitary action U_g^*U_h^* = U_{hg}^*, not U_{gh}^*; moreover, the sentence \"This follows from the fact that the orthogonal projection P is idempotent and has operator norm bounded by 1\" assumes the conclusion. A correct proof can be supplied by showing, via the definition of •, that (p•q)-lim_g U_g = Φ(q)Φ(p), so p^2=p implies Φ(p)^2=Φ(p), and a norm-bounded idempotent operator on Hilbert space is self-adjoint. Without such a proof, the left IP* conclusion of Theorem 5, and hence Theorem A, is unsupported.","section":"Section 2.2, Theorem 5 and preceding paragraph"},{"comment":"The step after obtaining B in W with R_{b,g,(1,...,k)}⊆A is too compressed. The line \"In accordance with Lemma 9 and using point 2 we can deduce that there exists B∈W such that for all b∈B we have R_{b,g,(2,...,k+1)}⊆A\" omits the coordinate-permutation argument, and the verification that for \\tilde B = B g_1^{-1} the shifted points R^{(1)}_g...R^{(j)}_g \\tilde b lie in A requires the coordinate identities \\tilde b g_j = b γ_2...γ_j, which are not shown. Since this implication feeds into the induction for Theorem B, the proof needs to be written out. Similarly, Theorems 19 and 20 are justified only as \"essentially a repetition\" of Theorems 11 and 12; the hypotheses differ in a substantive way (a van der Waerden family W replaces positive-density sets), so the reader needs a precise statement of the analogy or a full proof.","section":"Section 2.5, Theorem 18, proof of 2⇒3"}],"minor_comments":[{"comment":"In the definition of the topology on βG, the basis is written as {A : A⊆G}, but it should be {\\bar A : A⊆G} with \\bar A={p∈βG : A∈p}; the overline notation is missing.","section":"Section 2.1"},{"comment":"The permutation σ_k and its inverse are described inconsistently: with σ_k=(1,k+1,k,...,3,2), the inverse is not the identity (1,2,3,...,k+1). The intended permutation should be clarified.","section":"Lemma 9"},{"comment":"In part (2), the elements x and y (and h) belong to G^m, not to G; as written the quantifiers are formally wrong.","section":"Section 2.5, Theorem 18"},{"comment":"The statements omit the van der Waerden family W from their hypotheses even though their conclusions involve sets A_i∈W and their proofs use the van der Waerden property; the dependence on W should be stated explicitly.","section":"Theorems 19 and 20"},{"comment":"The text contains several typographical artifacts ('/greaterorequalslant', 'exsits', 'monochormatic', 'FC-grou ps' in the title) and an illustrative example in the introduction that is difficult to parse; these should be cleaned up.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main gap in Theorem 5 is real but appears fixable by a standard argument; I do not see a fundamental obstruction to the claimed results. The paper would also benefit from expanding the compressed proofs in Section 2.5. If the author can supply the missing projection proof and clarify the induction steps, I would support publication in this journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is real progress on a hard conjecture, and the main theorems are probably right, but the proof as written has a load-bearing gap that needs to be fixed before I'd trust it.\n\nThe genuinely new content is Theorem A—m=2 of the Bergelson–McCutcheon conjecture for all countable amenable groups, with the conclusion strengthened from central* to IP*—and Theorem B, the full m-dimensional version for FC-groups. The color-focusing lemma (Theorem 11) is a nice technique and looks reusable. The van der Waerden group framework in Section 2.5 is a sensible way to isolate what is needed for higher dimensions. The paper also engages honestly with the literature: no self-citation as support, the Bergelson–McCutcheon results are cited properly, and the open questions at the end are genuine.\n\nThe soft spot is Theorem 5. The paper needs P f = p-lim T_g f to be an orthogonal projection for every idempotent p, not just minimal ones. The cited Theorem 4 gives that only for minimal idempotents, and the paper's attempt to prove the general case contains a wrong computation: it claims U_g^*U_h^* = U_{gh}^*, while the correct identity is U_g^*U_h^* = U_{hg}^*. The conclusion is actually true—Φ(p) = WOT-p-lim U_g is a semigroup homomorphism, so idempotence follows—but the text doesn't say that. Since Theorem A and Theorem 6 and everything downstream depend on Theorem 5, this has to be sorted out.\n\nThe other weakness is compression in Section 2.5: Theorem 19 is \"essentially a repetition,\" Theorem 20 likewise, and the FC-group fact that left syndetic equals right syndetic is cited to MathOverflow. These are minor compared to the projection issue, but a referee should ask for them to be written out.\n\nThe overall strategy is sound, there is no circular reasoning, and the claimed theorems are new. I'd send this to a serious referee rather than desk reject, with instructions to demand a complete proof of Theorem 5 and expansion of the compressed steps. I'd probably wait to cite it until that's on the arXiv.","headline":"Real progress on Bergelson–McCutcheon, but Theorem A rides on an unproved projection lemma whose sketch is wrong; the result is likely true and deserves refereeing.","tokens_in":18480,"tokens_out":4537,"would_cite":false,"duration_ms":41605,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D10","20F24","43A07","37A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any finite coloring of the square of a countable amenable group, the set of shifts that yield a monochromatic right corner is a left IP* set.","keywords":["amenable groups","FC-groups","IP* sets","van der Waerden theorem","Ramsey theory","Stone-Čech compactification","monochromatic corners","ergodic Ramsey theory"],"falsifier":"If an idempotent ultrafilter $p$ and a unitary $G$-action could be exhibited for which the weak $p$-limit $P f = p\\text{-}\\lim_g U_g f$ is not an orthogonal projection (e.g., $P^2 \\neq P$), the IP-Poincaré lemma behind Theorem A would fail and the proof would collapse.","tokens_in":17431,"feed_emoji":"🎯","tokens_out":12432,"duration_ms":105704,"temperature":0.7,"pith_summary":"This paper proves a non-abelian van der Waerden theorem in the plane: for any countable discrete amenable group $G$ and any finite coloring of $G^2$, some color class contains a monochromatic right corner $\\{(x,y),(xg,y),(xg,yg)\\}$, and the set of shifts $g$ that work is left IP* — it meets every left IP set. This settles the $m=2$ case of the Bergelson–McCutcheon conjecture, upgrading the previously known central* conclusion to IP* in that case. The proof combines an IP-Poincaré recurrence statement for a single measure-preserving anti-action with a new color-focusing lemma. For FC-groups (groups whose conjugacy classes are finite), the same conclusion is proved for all dimensions $G^m$.","feed_headline":"Every finite coloring of an amenable group square hides a monochromatic corner","feed_subtitle":"The m=2 conjecture holds for every countable amenable group; FC-groups give every dimension.","key_machinery":"The load-bearing mechanism is a one-parameter IP-Poincaré recurrence theorem: for a measure-preserving $G$-anti-action on a Lebesgue space and any $A$ of positive measure, the set $\\{g : \\mu(A\\cap T_g^{-1}A)>0\\}$ is both right and left IP*. The proof obtains this through an arbitrary idempotent ultrafilter $p$ on $G$, writing $Pf = p\\text{-}\\lim_g T_g f$ and relying on $P$ being an orthogonal projection. Applying the Furstenberg correspondence principle turns this into a density statement on $G^2$: for a set $E\\subseteq G^2$ of positive upper density, $\\{g : \\bar d_\\Phi(E\\cap E(g^{-1},e))>0\\}$ is IP*. A color-focusing lemma then iterates that density input: from any IP set $X$ it builds nested IP sets and nested monochromatic positive-density sets, with shifts chosen inside the IP sets, so that eventually two same-color layers produce a monochromatic corner with shift in $X$. For FC-groups, the density family is replaced by left piecewise syndetic sets, which are shown to form a van der Waerden family; the same focusing scheme then works in every dimension $G^m$.","core_discovery":"The central discovery is Theorem A: if $G$ is countable, discrete, and amenable, then for every finite partition $G^2=\\bigcup_{j=1}^r C_j$ there is a color $j$ such that the set $\\{g\\in G : \\exists (x,y)\\in G^2 \\text{ with } \\{(x,y),(xg,y),(xg,yg)\\}\\subseteq C_j\\}$ is a left IP* set. A left IP* set is one that meets every left IP set, i.e. every set containing a decreasing finite-product set $\\{\\prod^\\downarrow_{n\\in F} x_n\\}$ of some sequence. A symmetric statement for left corners gives a right IP* set. The same conclusion holds for $G^m$ for every $m$ whenever $G$ is an FC-group, meaning all conjugacy classes are finite; this includes all abelian groups and examples such as $\\mathbb{Z}\\times Q_8$.","pith_inferences":["If the projection-property step for arbitrary idempotent ultrafilters is made fully rigorous, the same scheme should reach $G^3$: the paper isolates the missing density recurrence as the only obstruction.","A promising place to search for the boundary of the amenable-group result is nilpotent groups, where left and right syndetic sets differ; the FC-group proof fails exactly at that symmetry.","The paper's dichotomy suggests that if the full conjecture fails for some amenable group, the failure should appear in the van der Waerden family property, not in the color-focusing machinery."],"forward_implications":["For every left IP set $X\\subseteq G$ and every finite coloring of $G^2$, a monochromatic right corner occurs with shift $g\\in X$; this is exactly what IP* means.","The two-dimensional case of the Bergelson–McCutcheon conjecture is resolved, replacing the weaker central* conclusion by the stronger IP* one.","For every FC-group, including abelian groups, $\\mathbb{Z}\\times Q_8$, and restricted direct products of dihedral groups, the conjectured multidimensional statement holds for all $m$.","The color-focusing lemma works in all dimensions, so the only obstacle to the full conjecture for arbitrary amenable groups is an IP* density recurrence for intersections along more than one coordinate."],"supporting_citations":[{"why":"Supplies the Bergelson–McCutcheon central* theorem, the unitary-projection theorem for minimal idempotents that the paper adapts, and the conjecture being partially solved.","marker":"[6]"},{"why":"Gives the IP-van der Waerden theorem for abelian groups that serves as the model statement and the source of the IP* notion.","marker":"[8]"},{"why":"Provides the Stone-Čech algebra facts: IP sets are exactly sets contained in idempotents, IP* sets meet all idempotents, and product subsystems can be chosen inside subsemigroups, used throughout the color-focusing lemma.","marker":"[13]"},{"why":"Establishes that left and right syndetic sets coincide in FC-groups, the key fact used to show piecewise syndetic sets form a van der Waerden family.","marker":"[10]"},{"why":"Used to conclude that FC-groups are amenable, a hypothesis required by the van der Waerden-group theorem.","marker":"[15]"},{"why":"Provides the counterexample for infinitely generated free groups that delimits why the paper works with amenable groups.","marker":"[2]"}],"fun_headline_variants":["Amenable groups: every finite coloring of G^2 yields a monochromatic corner","FC-groups extend monochromatic corner theorem to all dimensions","Every finite coloring of an amenable group square has a monochromatic corner","Corner theorem for amenable groups: left IP* sets guarantee monochromatic triples","Van der Waerden type theorem: amenable groups and FC-groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the claim that weak limits through arbitrary idempotent ultrafilters are orthogonal projections; if some idempotent fails that, the upgrade from central* to IP* no longer follows from the presented argument.","fun_headline_variants_meta":{"raw":{"variants":["Amenable groups: every finite coloring of G^2 yields a monochromatic corner","FC-groups extend monochromatic corner theorem to all dimensions","Every finite coloring of an amenable group square has a monochromatic corner","Corner theorem for amenable groups: left IP* sets guarantee monochromatic triples","Van der Waerden type theorem: amenable groups and FC-groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000663,"raw_usage":{"total_tokens":2984,"prompt_tokens":858,"completion_tokens":2126,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":474,"completion_tokens_details":{"reasoning_tokens":2030}},"tokens_in":474,"tokens_out":2126,"duration_ms":14515,"temperature":1.0,"reasoning_tokens":2030,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:42:16.862960+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If an idempotent ultrafilter $p$ and a unitary $G$-action could be exhibited for which the weak $p$-limit $P f = p\\text{-}\\lim_g U_g f$ is not an orthogonal projection (e.g., $P^2 \\neq P$), the IP-Poincaré lemma behind Theorem A would fail and the proof would collapse.","supporting_citations":[{"cited_title":"Bergelson, and R","cited_arxiv_id":null,"evidence_quote":"Supplies the Bergelson–McCutcheon central* theorem, the unitary-projection theorem for minimal idempotents that the paper adapts, and the conjecture being partially solved."},{"cited_title":"Bergelson, Ergodic Ramsey Theory–an Update , 1996","cited_arxiv_id":null,"evidence_quote":"Gives the IP-van der Waerden theorem for abelian groups that serves as the model statement and the source of the IP* notion."},{"cited_title":"Hindman, and D","cited_arxiv_id":null,"evidence_quote":"Provides the Stone-Čech algebra facts: IP sets are exactly sets contained in idempotents, IP* sets meet all idempotents, and product subsystems can be chosen inside subsemigroups, used throughout the color-focusing lemma."},{"cited_title":"de Cornulier, Left syndeticity and right syndeticity in nilpotent group , MathOverflow, 2022","cited_arxiv_id":null,"evidence_quote":"Establishes that left and right syndetic sets coincide in FC-groups, the key fact used to show piecewise syndetic sets form a van der Waerden family."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used to conclude that FC-groups are amenable, a hypothesis required by the van der Waerden-group theorem."},{"cited_title":"Bergelson, and N","cited_arxiv_id":null,"evidence_quote":"Provides the counterexample for infinitely generated free groups that delimits why the paper works with amenable groups."}],"review_version":1}