Chromatic thresholds for linear equations and recurrence

Hong Liu; Ningyuan Yang; Shengtong Zhang; Zhuo Wu

REVIEW 3 major objections 2 minor 1 cited by

Chromatic thresholds for linear equations and recurrence

T0 review · 3 major / 2 minor · reviewed 2026-07-15 · grok-4.5

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

Pith's one-line read A linear equation has chromatic threshold zero exactly when it contains a zero-sum subcollection of three or more coefficients.

desk verdict Clean classification of when the chromatic threshold vanishes for linear equations, but only the abstract is available so the load-bearing lower bound cannot be checked. read the letter →

arxiv 2603.05490 v2 pith:ICRIHHPU submitted 2026-03-05 math.CO

Hong Liu , Zhuo Wu , Ningyuan Yang , Shengtong Zhang This is my paper · ORCID

classification math.CO MSC 05C1505C2511B3037B20

keywords chromaticthresholdlinearequationsCayleygraphsfinitefieldssolution-freesetsrecurrenceKneserBorsuk-Ulam

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 studies a chromatic analogue of classical Roth-type problems for a homogeneous linear equation L over a prime field. It measures L-solution-free sets A through the chromatic number of the Cayley graph Cay(F_p, A) and defines the chromatic threshold δ_χ(L) as the least density that forces every such free set to produce a Cayley graph of bounded chromatic number. The authors prove that this threshold is exactly zero if and only if L contains a subcollection of at least three coefficients that sum to zero. The proof rests on a quantitative lower bound for the chromatic numbers of certain Cayley graphs on vector spaces over F_p, obtained from a new Kneser-type graph that embeds into Z_p^n together with an equivariant Borsuk–Ulam argument. The classification resolves a question of Griesmer and yields, in every infinite discrete abelian group, a set that is topologically recurrent but not measurably recurrent.

What carries the argument

A new Kneser-type graph that admits a natural embedding into Z_p^n, combined with an equivariant Borsuk–Ulam-type argument, producing a quantitative chromatic lower bound for Cayley graphs generated by Hamming balls around the all-ones vector.

What would settle it

Produce a homogeneous linear equation with no zero-sum subcollection of three or more coefficients for which every solution-free set above some fixed positive density has bounded-chromatic Cayley graph, or produce an equation that does possess such a subcollection yet still admits arbitrarily sparse free sets whose Cayley graphs have unbounded chromatic number.

Watch

Extended reading notes

Core claim

The chromatic threshold δ_χ(L) of a homogeneous linear equation L over F_p equals zero if and only if L contains a zero-sum subcollection of at least three coefficients. Equivalently, among L-solution-free sets A the Cayley graph Cay(F_p, A) has unbounded chromatic number at every positive density precisely when no such subcollection exists.

Load-bearing premise

The quantitative chromatic lower bound for Cayley graphs generated by Hamming balls around the all-ones vector is strong enough to force unbounded chromatic number at every positive density whenever the equation has no zero-sum subcollection of size three or larger.

Editorial extensions

If this is right

If L admits a zero-sum subcollection of size at least three, every positive-density L-free set yields a Cayley graph of bounded chromatic number.
If L admits no such subcollection, there exist L-free sets of arbitrarily small positive density whose Cayley graphs have unbounded chromatic number.
Every infinite discrete abelian group contains a set that is topologically recurrent but not measurably recurrent.
Griesmer’s question on the existence of certain recurrent sets is settled by the classification.

Reading between the lines

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

The same zero-sum criterion may control chromatic thresholds for systems of several linear equations rather than a single equation.
The Kneser-type construction may supply chromatic lower bounds for other families of Cayley graphs generated by metric balls.
Analogous thresholds over the integers could be approachable by lifting the finite-field constructions via Fourier analysis or transfer principles.

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 to know: this abstract claims an exact classification of when the chromatic threshold δ_χ(L) vanishes for a homogeneous linear equation L over F_p—namely, iff L has a zero-sum subcollection of at least three coefficients—and packages it with a new Kneser-type construction plus a uniform topological-but-not-measurable recurrence result for every infinite discrete abelian group.

What looks new and useful is the framing itself. Chromatic thresholds for graphs are classical; transplanting the idea to Cayley graphs of L-solution-free sets and getting a clean combinatorial criterion is a natural and apparently original move. The claimed resolution of Griesmer’s question and the uniform extension of the Kříž–Ruzsa phenomenon are both worth having if the proofs hold. The abstract is precise about the statement and about the technical ingredients (Hamming-ball generators on Z_p^n, a new Kneser-type graph with an embedding, equivariant Borsuk–Ulam).

The soft spot is exactly the one the stress-test flags, and it is load-bearing: the “only if” direction rests on a quantitative chromatic lower bound that we cannot inspect. Without the embedding, the equivariance, or the density-independence of the bound, we cannot confirm that unbounded chromatic number really holds for every positive density when no zero-sum triple of coefficients exists. Everything else—circularity, free parameters, invented entities—looks clean; this is pure classification work. The limitation is simply that we have only the abstract.

This is for people who work on additive combinatorics, chromatic thresholds, or recurrence hierarchies. A serious referee should see the full manuscript; the claim is sharp enough and the surrounding literature is mature enough that desk rejection would be the wrong call. I would bring it to reading group once the proofs are public, and I would cite the classification if it survives. Send it to peer review.

Referee Report

3 major / 2 minor

Summary. The manuscript studies a chromatic analogue of Roth-type problems for homogeneous linear equations L over F_p. For L-solution-free sets A ⊆ F_p it considers the chromatic number of the Cayley graph Cay(F_p,A) and introduces the chromatic threshold δ_χ(L), the infimal density guaranteeing that this chromatic number remains bounded. The main claim is an exact classification: δ_χ(L)=0 if and only if L contains a zero-sum subcollection of at least three coefficients. The “only if” direction is obtained from a quantitative chromatic lower bound for Cayley graphs on Z_p^n generated by Hamming balls around the all-ones vector, via a newly introduced Kneser-type graph that embeds into Z_p^n together with an equivariant Borsuk–Ulam argument. As consequences the authors resolve a question of Griesmer and relate the classification to measurable, topological and Bohr recurrence, showing in particular that every infinite discrete abelian group admits a set that is topologically recurrent but not measurably recurrent.

Significance. If the classification and the supporting lower bound hold, the paper supplies a clean, parameter-free dichotomy for when L-solution-free sets of arbitrarily small positive density can force unbounded chromatic number of the associated Cayley graphs. The resolution of Griesmer’s question and the extension of the Kříž–Ruzsa examples to every infinite discrete abelian group would be substantial contributions at the interface of combinatorial number theory, topological combinatorics and recurrence theory. The introduction of a new Kneser-type graph with a natural embedding into Z_p^n is potentially of independent interest. These strengths, however, rest entirely on technical ingredients that cannot be inspected from the abstract alone.

major comments (3)

[Abstract (key ingredient paragraph)] The central classification δ_χ(L)=0 ⇔ L has a zero-sum subcollection of size ≥3 is load-bearing for the whole paper. The “only if” direction (and the resolution of Griesmer’s question) depends on a quantitative chromatic lower bound for Cayley graphs Cay(Z_p^n, Hamming ball about the all-ones vector). That bound is obtained from a newly introduced Kneser-type graph plus an equivariant Borsuk–Ulam argument. With only the abstract available, neither the embedding, the equivariance, the resulting numerical lower bound, nor its claimed density-independence can be verified. Until the full argument is supplied and checked, the classification remains unconfirmed.
[Abstract (classification statement and key ingredient)] The abstract asserts that the new lower bound is strong enough to force unbounded chromatic number for every positive density whenever L has no zero-sum subcollection of size ≥3. This density-independence claim is precisely the ingredient needed to conclude δ_χ(L)>0 in that case. Without the quantitative estimate (or even a sketch of how the Borsuk–Ulam degree or index produces a bound independent of density), it is impossible to assess whether the lower bound actually reaches the threshold required by the classification.
[Abstract (final paragraph)] The recurrence-theoretic consequences (topological but not measurable recurrence on every infinite discrete abelian group) are presented as flowing from the same classification. If the chromatic lower bound is weaker than claimed, the extension beyond the classical Kříž–Ruzsa examples may fail or require additional hypotheses. The logical dependence should be made explicit once the full text is available.

minor comments (2)

[Abstract] The abstract is clear and well-structured, but the notation for the chromatic threshold δ_χ(L) and for the equation L itself should be fixed consistently once the full manuscript is supplied (script L versus calligraphic L, etc.).
[Abstract] A precise statement of Griesmer’s question, even in one sentence, would help the reader locate the contribution; the abstract only says the question is resolved.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detectable from the abstract: pure classification theorem with no fitted inputs or definitional reductions.

full rationale

Only the abstract is available. It states a clean existence/classification result: δ_χ(L)=0 iff the homogeneous linear equation L contains a zero-sum subcollection of at least three coefficients. The claimed ingredients (a new Kneser-type graph embedding into Z_p^n, an equivariant Borsuk–Ulam argument yielding a quantitative chromatic lower bound for Hamming-ball Cayley graphs, and consequences for Griesmer’s question and recurrence hierarchies) are presented as constructions and proofs, not as fitted parameters, self-definitions, or renamings of the target statement. There is no parameter fitting, no “prediction” that reduces to a fitted constant by construction, no uniqueness theorem imported from the authors’ prior work that forces the classification, and no self-citation chain that substitutes for the argument. Per the hard rules, circularity may be claimed only when a specific reduction can be quoted and exhibited; none appears in the available text. The skeptic’s concern that the lower-bound ingredient cannot be inspected is a verification/correctness issue, not circularity. Score 0 is therefore the honest finding.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

Pure mathematical paper; no numerical free parameters. Background axioms are standard finite-field arithmetic, Cayley-graph chromatic number, and classical notions of measurable/topological/Bohr recurrence. The paper invents the chromatic threshold δ_χ(L) as a definition and a new Kneser-type graph as a technical tool; the latter is justified by the claimed embedding and Borsuk–Ulam argument rather than by external empirical evidence.

assumptions (4)

standard math Standard arithmetic and linear algebra over finite fields F_p and vector spaces Z_p^n
Used throughout for equations, Cayley graphs, and Hamming balls.
standard math Existence and basic properties of equivariant Borsuk–Ulam-type theorems for the relevant group actions
Invoked as the topological engine for the chromatic lower bound on the new Kneser-type graph.
domain assumption Standard definitions of measurable, topological, and Bohr recurrence for discrete abelian groups
Needed to state and compare the recurrence hierarchy and to extend Kříž–Ruzsa examples.
domain assumption That L-solution-free sets and the chromatic number of Cay(F_p,A) are the correct objects for a chromatic Roth-type threshold
Motivates the definition of δ_χ(L); standard in the chromatic-threshold program but a modeling choice relative to pure density Roth theorems.

invented entities (2)

chromatic threshold δ_χ(L)
purpose: Minimum density guaranteeing bounded chromatic number of Cay(F_p,A) among L-solution-free sets A
Definitional object of the paper; independent evidence is the classification theorem itself once proved.
new Kneser-type graph with natural embedding into Z_p^n
purpose: Provide quantitative chromatic lower bounds for Cayley graphs generated by Hamming balls around the all-ones vector
Technical construction introduced to power the lower-bound half of the classification and to resolve Griesmer’s question; falsifiability is internal to the combinatorial argument.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Chromatic thresholds for linear equations and recurrence." pith.science (2026). https://pith.science/paper/ICRIHHPU

@misc{pith2026260305490,
  author       = {Pith},
  title        = {Pith review of: Chromatic thresholds for linear equations and recurrence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICRIHHPU}},
  note         = {Machine review of arXiv:2603.05490}
}

abstract

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study $\mathcal L$-solution-free sets $A\subseteq \mathbb F_p$ through the chromatic number of the Cayley graph $\mathsf{Cay}(\mathbb F_p,A)$. We introduce the \emph{chromatic threshold} $\delta_\chi(\mathcal L)$, the minimum density that guarantees bounded chromatic number of $\mathsf{Cay}(\mathbb F_p,A)$ among all $\mathcal L$-solution-free sets $A$, and determine exactly when $\delta_\chi(\mathcal L)=0$. We prove that $\delta_\chi(\mathcal L)=0$ if and only if $\mathcal L$ contains a zero-sum subcollection of at least three coefficients. A key ingredient is a quantitative chromatic lower bound for Cayley graphs on $\mathbb Z_p^n$ generated by Hamming balls around the all-ones vector. This is obtained by introducing a new Kneser-type graph that admits a natural embedding into $\mathbb Z_p^n$, together with an equivariant Borsuk--Ulam type argument. As a consequence, we resolve a question of Griesmer. We further relate our classification to the hierarchy of measurable, topological, and Bohr recurrence. In particular, we show that every infinite discrete abelian group admits a set that is topological recurrent but not measurable recurrent, extending the seminal examples of K\v{r}\'i\v{z} and Ruzsa.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Linear equations and chromatic thresholds in $B_h$ sets
math.CO 2026-06 unverdicted novelty 6.0 of 10

Sparse Roth-type results hold in near-maximal B_h sets; Sidon sets free of equations with five-coefficient zero-sums are small or have bounded-chromatic Cayley graphs, with constructions for the four-coefficient case.

Pith tools

Reviewed July 15, 2026 · model on record in the stance chip above.