REVIEW 2 major objections 3 minor 2 cited by
Asymptotic structure. IV. A counterexample to the weak coarse Menger conjecture
T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read This paper proves the weak coarse Menger conjecture false: for every pair of constants (ell,m), there is a graph with no three S-T paths pairwise at distance at least 3, yet no set of at most m vertices meets every S-T path within distance
desk verdict The abstract claims a decisive counterexample to the weak coarse Menger conjecture; the result is significant and plausible, but the proof is not visible, so the referee should focus on the amplification step. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing device is a parameterized family of graphs G(ell,m) with distinguished vertex sets S,T, built by amplifying the authors' earlier c=k=3 counterexample. The construction preserves the blocking property—absence of three S-T paths pairwise at distance at least 3—while forcing any radius-ell hitting set to grow beyond m. The distance between paths is measured as the minimum graph distance between a vertex of one path and a vertex of the other; the proof's work is showing these two coarse parameters, separator size and separator radius, can be driven apart without limit.
What would settle it
For one concrete pair (ell,m), inspect the constructed graph: if three S-T paths pairwise at distance at least 3 exist, the main claim fails; if a set X of size at most m has every S-T path pass within distance ell of X, the main claim also fails. Verifying either for any single pair would overturn the universal counterexample.
Extended reading notes
Core claim
The central claim is a universal counterexample. The paper proves: for every choice of integers ell and m, there exists a graph G with designated vertex sets S and T such that (1) G has no three S-T paths that are pairwise at graph distance at least 3, and (2) for every X subset of V(G) with |X| <= m, there is at least one S-T path whose vertices all lie at distance greater than ell from X. Since condition (1) matches the hypothesis of the weak coarse Menger conjecture for k=3, c=3, and condition (2) contradicts the conjectured conclusion for any proposed constants, the conjecture cannot hold for any pair of constants. The earlier counterexample for the stronger statement is upgraded so that
Load-bearing premise
The earlier three-path counterexample is valid and can be amplified without creating three S-T paths that are pairwise at least 3 apart; if the amplification accidentally produces such a triple, the universal statement collapses.
Editorial extensions
If this is right
- The weak coarse Menger conjecture is false in its full generality: no function of k and c bounds the separator size and radius, even when k=c=3.
- Any positive coarse Menger theorem must restrict the graph class, for example by excluding a fixed minor, or must allow the separator size and radius to be tied to the graph rather than to k and c alone.
- The obstruction is already present with exactly three paths, so the failure is not a large-k effect; it is a local combinatorial phenomenon.
- The problem shifts from calibrating constants to identifying which families of graphs admit any coarse Menger-type bound at all.
Reading between the lines
- If the amplification used here is a genuine blow-up of the earlier three-path gadget, the same construction would likely rule out coarse Menger bounds in any graph class closed under that blow-up; candidate positive classes should be checked for whether they contain the amplified gadget.
- The theorem suggests the separator size and radius can be traded off against each other with no finite bound on the ratio; a natural next step is to measure how fast the minimal separator size grows as a function of ell for the constructed family.
- It may be possible to make the construction bounded-degree or minor-minimal; if so, even fairly tame graph classes would escape the conjecture, steering the search for positive results toward hypotheses that forbid the construction outright.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper, as represented by the abstract, claims to disprove the weak coarse Menger conjecture. It states that for all natural numbers ℓ and m, there is a graph G and vertex sets S,T such that (a) there do not exist three S–T paths pairwise at distance at least three, and (b) there is no X ⊆ V(G) with |X| ≤ m such that every S–T path passes within distance at most ℓ of X. This is presented as an upgrade of the authors' earlier counterexample to the natural coarse Menger analogue for c=k=3. The abstract is the only material provided for this review; the construction and proof are not included.
Significance. If the theorem is correct, it settles a prominent open question in coarse graph theory by showing that even the weak coarse Menger conjecture fails in the strongest sense: for c=k=3, no pair of constants (for the hitting-set size and the radius) can work. The statement is precise, concrete, and falsifiable, and the result would be a substantial negative contribution to the field. However, the significance is conditional: without the full construction and proof, the claim cannot be verified, and the current reviewable material does not establish the result.
major comments (2)
- [Abstract / Full text (not available)] The manuscript as reviewed consists only of the abstract. The central theorem is stated, but the construction of G and S,T and the proof of its properties are entirely omitted. For a claim that refutes a widely believed conjecture, the absence of a verifiable proof is a load-bearing omission, not a presentation issue. I cannot assess the soundness of the result from the submitted material.
- [Abstract, 'upgrade' step] The claimed counterexample upgrades the authors' earlier c=k=3 counterexample to defeat all choices of ℓ and m. The critical step must be an amplification that preserves the invariant 'there do not exist three S–T paths pairwise at distance at least three' while forcing every set X with |X|≤m to fail to ℓ-hit all S–T paths. The abstract gives no information about how this is achieved. This is exactly where a subtle error could arise—for instance, if the amplification adds fan-like structures or long connections that create three pairwise far paths. The full proof must demonstrate this invariant explicitly; without it, the main claim is unsupported.
minor comments (3)
- [Abstract] The abstract names the conjecture proposers but gives no citations. The full paper should include references to Georgakopoulos–Papasoglu and Albrechtsen–Huynh–Jacobs–Knappe–Wollan, as well as to the authors' own earlier paper containing the base counterexample.
- [Abstract] The phrase 'no pair of constants work' is then formalized as 'for all ℓ,m'; the intended quantifier structure is clear, but the wording could be tightened to avoid ambiguity about whether ℓ and m range over all positive integers.
- [General] If the full paper indeed contains Sections 2–3 with the construction, the abstract-only review format is insufficient for a serious journal. The authors should be asked to submit the complete manuscript for review.
Circularity Check
No circularity found in the abstract; the new result is a strict strengthening of prior work, not a restatement.
full rationale
The abstract's derivation chain is: (i) prior work by the same authors produced a counterexample to the natural |X|<k version with c=k=3; (ii) the present paper upgrades this to show that for every ell,m, no pair of constants works. This is a strict logical strengthening: the earlier statement asserts existence of one bad pair of constants, the new statement quantifies over all ell,m. No equation in the abstract defines the new graph in terms of the target conclusion, no fitted constants are renamed as predictions, and no uniqueness theorem is imported. The only reliance on earlier work is the use of a cited prior theorem as a base construction; that theorem is weaker than the new result and has its own proof, so the self-citation is not circular. The skeptics' concern (that the amplification might accidentally create three pairwise distance-3 S-T paths) is a mathematical correctness risk about invariant preservation, not a circular derivation. No exhibited reduction of the conclusion to its inputs appears in the abstract. Thus the circularity score is 0.
Assumptions & free parameters
assumptions (2)
- domain assumption The earlier counterexample with c=3,k=3 from the authors' previous paper is correct and can serve as a base construction.
- domain assumption The standard definitions of coarse graph theory (distance between paths, S-T paths, closeness to X) are those used in the conjecture being refuted.
Cite this review
Pith. "Pith review of Asymptotic structure. IV. A counterexample to the weak coarse Menger conjecture." pith.science (2026). https://pith.science/paper/QQX4FX3J
@misc{pith2026250814332,
author = {Pith},
title = {Pith review of: Asymptotic structure. IV. A counterexample to the weak coarse Menger conjecture},
year = {2026},
howpublished = {\url{https://pith.science/paper/QQX4FX3J}},
note = {Machine review of arXiv:2508.14332}
}
abstract
Coarse graph theory concerns finding 'coarse' analogues of graph theory theorems, replacing disjointness with being far apart. One of the most interesting open questions is to find a coarse analogue of Menger's theorem, which characterizes when there are $k$ vertex-disjoint paths between two given sets $S,T$ of vertices of a graph. We showed in an earlier paper that the most natural such analogue is false, but a weaker statement remained as a popular open question. Here we show that the weaker statement is also false. More exactly, suppose that $S,T$ are sets of vertices of a graph $G$, and there do not exist $k$ paths between $S,T$, pairwise at distance at least $c$. To make an analogue of Menger's theorem, one would like to prove that there must be a small set $X\subseteq V(G)$ such that every $S-T$ path of $G$ passes close to a member of $X$: but how small and how close? In view of Menger's theorem, one would hope for $|X|<k$ and 'close' some function of $k,c$ (and indeed, this was conjectured by Georgakopoulos and Papasoglu, and independently, by Albrechtsen, Huynh, Jacobs, Knappe and Wollan); but we showed that this is false, even if $c=3$ and $k=3$. Here we upgrade the counterexample: we show that, even if $c=k=3$, no pair of constants (for 'small' and 'close') work. For all $\ell, m$, there is a graph $G$ and $S,T\subseteq V(G)$, such that there do not exist three $S-T$ paths pairwise with distance at least three, and yet there is no $X$ with $|X|\le m$ such that every $S-T$ path passes within distance at most $\ell$ of $X$.
Forward citations
Cited by 2 Pith papers
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Every connected graph has a tree-decomposition with bounded-diameter adhesion sets and coarsely inseparable bags, yielding a metric analogue of the block-cutvertex tree.
Reviewed August 5, 2026 · model on record in the stance chip above.
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