Van der Waerden type theorem for amenable groups and FC-groups

Emilio Parini

REVIEW 2 major objections 5 minor 18 references

Van der Waerden type theorem for amenable groups and FC-groups

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.15987 v1 pith:C74ACCBT submitted 2024-11-24 math.GR math.CO

Emilio Parini This is my paper · ORCID

classification math.GRmath.CO MSC 05D1020F2443A0737A15

keywords amenablegroupsFC-groupsIP*setsvanderWaerdentheoremRamseytheoryStone-Čechcompactificationmonochromaticcornersergodic

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

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 $G^m$.

What carries the argument

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 $G^m$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $\mathbb{Z}\times Q_8$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

The 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.

The 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.

The 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 2.2, Theorem 5 and preceding paragraph] 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 2.5, Theorem 18, proof of 2⇒3] 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.

minor comments (5)

[Section 2.1] 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.
[Lemma 9] 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 2.5, Theorem 18] In part (2), the elements x and y (and h) belong to G^m, not to G; as written the quantifiers are formally wrong.
[Theorems 19 and 20] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper derives its results from external prior theorems and internal lemmas; the projection gap in §2.2 is a proof gap, not circular reasoning.

full rationale

The derivation chain is not circular. Theorem A is obtained from Theorem 13 and Theorem 12; Theorem 13 is derived from Theorem 6, which in turn follows from Theorem 5 and the Furstenberg correspondence principle. Theorem 5 invokes an external result, Theorem 4 of Bergelson–McCutcheon, and attempts to extend it from minimal idempotents to arbitrary idempotents. This extension is not framed as a prediction or as a consequence of the theorem being proved; it is an independent lemma, even if the supplied proof sketch is incomplete. No fitted parameters are involved, and no combinatorial conclusion is assumed in the hypotheses of the lemmas that establish it. The color-focusing argument in Theorem 11 is a genuine iterative construction from density assumptions, not a restatement of the target IP* property. For FC-groups, Theorem B is supported by the van der Waerden group framework, with Theorem 22 using the standard external fact that left and right syndetic sets coincide in FC-groups. The equivalences in Theorem 18 are proved back-and-forth rather than posited. The one genuinely questionable step is the claim that the weak p-limit P for an arbitrary idempotent p is an orthogonal projection; the paper's justification for P being self-adjoint appears to presuppose idempotence. That is a correctness concern, not a circular dependence of the theorem on its own conclusion: the projection property is not extracted from the IP* statement, nor is any quantity defined in terms of the target set. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on no fitted parameters or invented entities. It relies on standard ultrafilter, Følner, and correspondence techniques, plus a new projection lemma that is only sketched. The FC-group result uses known facts about FC-groups and syndetic sets.

assumptions (5)

standard math Stone-Cech compactification ultrafilter characterizations: a set is left IP iff it belongs to some idempotent, and left IP* iff it belongs to every idempotent.
Used throughout Section 2.1 and Theorem 5 to translate analytic limits into IP* largeness.
domain assumption Every countable amenable group admits right Følner sequences, and the associated right upper density is right invariant.
Introduced in Section 2.2 and underpins Theorem 6 and the color-focusing argument.
domain assumption Furstenberg correspondence principle for amenable group actions on G^2.
Invoked in Theorem 6 to convert a positive upper density set E into a measure-preserving system; cited to Bergelson-McCutcheon [6, Proposition 4.1].
ad hoc to paper For an arbitrary idempotent ultrafilter p, the weak p-limit P of the unitary anti-action is an orthogonal projection.
This extends [6, Theorem 2.4], which assumes minimal idempotents. The paper gives a sketch via P^2=P and norm bound but does not fully justify the arbitrary idempotent case; it is load-bearing for upgrading central* to IP* in Theorem 5.
domain assumption In FC-groups, left syndetic sets are right syndetic, and FC-groups are amenable via Paulsen's theorem.
Used in Theorem 22 to show that the return set L_A is right syndetic, giving the van der Waerden property for the family of left piecewise syndetic sets.

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Cite this review

Pith. "Pith review of Van der Waerden type theorem for amenable groups and FC-groups." pith.science (2026). https://pith.science/paper/C74ACCBT

@misc{pith2026241115987,
  author       = {Pith},
  title        = {Pith review of: Van der Waerden type theorem for amenable groups and FC-groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C74ACCBT}},
  note         = {Machine review of arXiv:2411.15987}
}

abstract

We prove that for a discrete, countable, and amenable group $G$, if the direct product $G^2=G \times G$ is finitely colored then $\{ g \in G : \text{exists } (x,y) \in G^2 \text{ such that } \{ (x,y),(xg,y),(xg,yg)\} \text{ is monochromatic} \}$, is left IP$^{\ast}$. This partially solves a conjecture of V. Bergelson and R. McCutcheon. Moreover, we prove that the result holds for $G^m$ if $G$ is an FC-group, i.e., all conjugacy classes of $G$ are finite.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Austin, Non-conventional ergodic averages for several commuting a ctions of an amenable group , Journal d’Analyse Mathématique, 130 (2013), 243–274

T. Austin, Non-conventional ergodic averages for several commuting a ctions of an amenable group , Journal d’Analyse Mathématique, 130 (2013), 243–274. URL

work page 2013

[2] [2]

Bergelson, and N

V. Bergelson, and N. Hindman, Some topological semicommutative van der Waerden type theorems and their combinatorial consequences , J. London Math. Soc. (2), 45 (1992), no. 3, 385–403. DOI

work page 1992

[3] [3]

Bergelson, N

V. Bergelson, N. Hindman, and R. McCutcheon, Notions of size and combinatorial properties of quotient sets in semigroups , Proceedings of the 1998 Topology and Dynamics Conference ( Fairfax, V A), Topology Proc., 23 (1998), 23–60

work page 1998

[4] [4]

Bergelson, J

V. Bergelson, J. Huryn, R. Raghavan, Discordant sets and ergodic Ramsey theory , Involve, 15 (2022), no. 1, 89–130. DOI

work page 2022

[5] [5]

Bergelson, J

V. Bergelson, J. H. Johnson, and J. Moreira, New polynomial and multidimensional extensions of classical partition results , J. Comb. Theory A, 147 (2015), 119–154. URL

work page 2015

[6] [6]

Bergelson, and R

V. Bergelson, and R. McCutcheon, Central sets and a non-commutative Roth theorem , Amer. J. Math., 129 (2007), no. 5, 1251–1275. DOI

work page 2007

[7] [7]

Bergelson, R

V. Bergelson, R. McCutcheon, and Q. Zhang, A Roth theorem for amenable groups , Amer. J. Math., 119 (1997), no. 6, 1173–1211. URL

work page 1997

[8] [8]

Bergelson, Ergodic Ramsey Theory–an Update , 1996

V. Bergelson, Ergodic Ramsey Theory–an Update , 1996. URL

work page 1996