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Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs

T0 review · 2 major / 6 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Above density 1/3, a sharp vertex-degree curve forces tight Hamilton cycles in uniformly dense 3-graphs; the codegree diagonal threshold is about 0.3177, not 1/4.

desk verdict Sharp thresholds that settle two named problems, with solid internal math that routes through one recent external embedding black box. read the letter →

arxiv 2607.23748 v1 pith:7K5JGZWJ submitted 2026-07-26 math.CO

Hao Lin , Guanghui Wang , Wenling Zhou This is my paper
classification math.CO MSC 05C6505C4505C35
keywords tightHamiltoncyclesuniformlydensehypergraphsminimumvertexdegreecodegreelinearquasirandomnessframeworksdominantcomponents
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

The paper asks when a 3-uniform hypergraph that is only mildly quasirandom—edges are spread roughly evenly—must contain a tight Hamilton cycle, provided its local degrees are not too small. For minimum vertex degree it gives the exact trade-off curve above density 1/3: density d together with normalized degree larger than f(d) is enough, and f(d) is tight. In particular the diagonal point (1/3,1/3) works, settling an open problem. For minimum codegree the diagonal threshold is the larger number κ≈0.3177 solving κ=(1−κ)³, so density and codegree both just above 1/4 do not force the cycle. Both proofs reduce Hamiltonicity to finding one dominant tight component that supplies a local Hamilton framework; the two degree regimes need different component lemmas.

What carries the argument

A Hamilton-framework reduction: almost every large induced subgraph must contain a spanning tight component that carries a perfect fractional matching, an aperiodic closed walk, and consistency across overlapping subgraphs; that component is located by two distinct dominant-component lemmas (weighted monochromatic-clique selection for vertex degree; stability around κ plus cherry inequalities for codegree).

What would settle it

Exhibit an infinite family of (n,d,μ)-dense 3-graphs with minimum vertex degree above f(d) (or codegree above κ) that still lack a tight Hamilton cycle, or show that the Lang–Sanhueza-Matamala embedding fails at the error scales used here.

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Extended reading notes

Core claim

In (n,d,μ)-dense 3-graphs, density d>1/3 and minimum vertex degree strictly above f(d)=(1−√((4d−1)/3))/2 force a tight Hamilton cycle, and the bound is asymptotically sharp; on the codegree side the sharp diagonal is exactly (κ,κ) with κ the unique root of κ=(1−κ)³≈0.3177, so the previously conjectured 1/4 threshold is false.

Load-bearing premise

The argument treats an external embedding theorem as a black box: once almost every medium-sized subset carries a local Hamilton framework, the whole graph is automatically Hamiltonian.

Editorial extensions

If this is right

  • The diagonal vertex-degree problem above density 1/3 is settled: any α>1/3 works once d>1/3.
  • The codegree conjecture that density and codegree >1/4 suffice is false; the correct diagonal constant is κ≈0.3177.
  • The same dominant-component lemmas supply structural information usable for other spanning problems in linearly quasirandom 3-graphs.
  • Below density 1/3 the threshold functions h1(d) and h2(d) remain open and are now the natural next targets.

Reading between the lines

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

  • The jump from 1/4 to κ on the codegree diagonal suggests that space barriers and component separation, not mere density, govern the obstruction once pairs rather than vertices are controlled.
  • Because the weighted-clique step needs monochromatic triangle weight >1/3, the method itself explains why density 1/3 is a natural wall for the vertex-degree argument.
  • Independent determination of the codegree curve above 1/3 (h2(d)=f(d)²) together with this diagonal result nearly completes the picture except in the sparse regime d≤1/3.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper studies tight Hamilton cycles in (n,d,μ)-dense 3-graphs under minimum-degree conditions. Theorem 1.1 shows that for every d∈(1/3,1] and α>f(d)=(1−√((4d−1)/3))/2, uniform density together with δ₁≥α\binom{n−1}{2} forces a tight Hamilton cycle, and §3.1 gives matching constructions showing this boundary is sharp; the diagonal consequence (Corollary 1.2) answers Problem 8.3(i) of Araújo–Piga–Schacht and confirms Conjecture 8.1 of Han–Shu–Wang. For the codegree condition, Theorem 1.3 constructs (n,κ,μ)-dense 3-graphs with δ₂≥(κ−ε)n and no tight Hamilton cycle, where κ=(1−κ)³≈0.3177>1/4, thereby disproving Conjecture 8.2 of Han–Shu–Wang and answering Problem 8.3(ii) negatively; Theorem 1.4 shows (κ,κ) is the sharp diagonal threshold. The upper-bound proofs use a common reduction to the Hamilton-framework embedding theorem of Lang–Sanhueza-Matamala via inheritance lemmas, with the framework conditions supplied by two distinct dominant tight-component lemmas: in the vertex-degree case via a component coloring, Kruskal–Katona, multicolor regularity, and a new weighted monochromatic-clique theorem; in the codegree case via a stability analysis around κ=(1−κ)³ and a weighted two-color cherry-counting lemma.

Significance. If correct, this is a strong result that settles two open problems from Araújo–Piga–Schacht and Han–Shu–Wang, one positively and one negatively, and in the vertex-degree case determines the entire sharp boundary curve α=f(d) for d>1/3 rather than only the diagonal. The negative codegree answer is quantitatively interesting: the diagonal threshold κ=(1−κ)³≈0.3177 strictly exceeds 1/4, so density and codegree above 1/4 provably do not suffice, and the matching construction in §3.2 is new and clean. Beyond the headline theorems, the paper contributes reusable machinery: the weighted monochromatic-clique theorem (Lemma 5.3) and its robust form (Theorem 5.4), the two-color structural theorem (Theorem 5.8), the sharp cherry lemma (Lemma 5.9, sharp at q=1/4), and a dominant-component method that avoids absorption. The sharpness constructions are explicit and the concentration arithmetic is verified in full, so the thresholds are derived from matching extremal examples rather than assumed. The combination of new internal machinery with the recent Hamilton-framework embedding technology is a natural and apparently effective way to bypass the connecting-ends obstacle identified in earlier的工作

major comments (2)
  1. [§2.1, Lemma 2.2; §4.3] Both main theorems route through Lemma 2.2, stated as 'the case k=t=3 of [30, Theorem 3.5]' (Lang–Sanhueza-Matamala, arXiv:2412.14891v2, revised March 2026, apparently unpublished), together with the inheritance statements Lemma 2.3 ([29, Lemma 8.1], arXiv:2308.12281v3, revised May 2026) and Lemma 2.4 ([30, Lemma 4.3]). If the specialization is inaccurate in any framework axiom — e.g. if [30]'s framework conditions differ from (F1)-(F4) of Definition 2.1, or if the quantitative hypothesis of [30, Thm 3.5] is weaker/stronger than the 'at least (1-1/s^2) of the s-sets through every 6-set' formulation used here — the deductions in §4.3 have no fallback. Since both cited works are very recent, revised preprints, the manuscript should include a short appendix or an expanded remark that (i) derives the k=t=3 statement from [30, Thm 3.5] explicitly (identifying the parameters and confirming the
  2. [§7.2, proof of Lemma 4.3] Lemma 4.3 is load-bearing for (F4) in the codegree case (Lemma 4.5), but its proof in §7.2 ends with the truncated sentence 'contradicting.' with no object. From context the intended contradiction is with Lemma 5.9 (with q' in place of q, on the reduced red weight system and its complement), but the argument as printed also skips a step: Lemma 5.9 gives a positive proportion βr_j^2/2 of bad pairs, whereas the proof establishes the common-neighbor lower bounds only on 'all but o(r_j^2)' pairs, so one must verify that o(r_j^2) < βr_j^2/2 for large j — this is fine but should be written out, and the missing reference restored. As printed, the final step of the proof is not checkable.
minor comments (6)
  1. [§5.2, opening paragraph] The sentence 'We prove Lemmas 4.2 and 4.3. Throughout this section, κ is...' appears at the head of §5.2, but §5.2 contains the weighted-coloring auxiliary results (Lemmas 5.5-5.9); Lemmas 4.2 and 4.3 are proved in §7. The sentence presumably belongs at the start of §7.
  2. [notation throughout §§5-7] The symbol q is used for the cluster size in §6, for the cherry-weight threshold in Lemma 5.9 and §7, and for the common-neighbor parameter in Lemma 2.12; m is used both for the number of clusters (§6) and as the induction variable in Lemma 5.3. Consider renaming to avoid confusion.
  3. [§4.1, Lemma 4.3 statement] In the statement of Lemma 4.3, the quantification 'there exist ε>0 and s0... Let R and B be... Then no such pair (R,B) exists' is slightly awkward; the conclusion is that the hypotheses are mutually inconsistent. A phrase such as 'no such pair exists' directly after the hypotheses would read better.
  4. [§4.3] Lemma 2.2 requires s≥6 and a statement for every 6-set R; a sentence explaining why r=6 (and not a larger anchor) suffices in the applications of Lemmas 2.3-2.4 in §4.3 would help the reader follow the constant hierarchy d0,α0 → μ',s0 → s → μ → n0.
  5. [References] Please double-check reference data: [17] is listed as J. Combin. Theory Ser. B 177:1-30, 2026; [29] and [30] are cited only as preprints with 2026 revision dates. If any of these have since appeared, updated citations (and, for [30], a stable statement number for Theorem 3.5) should be given.
  6. [§1.3, Corollary 1.2] The diagonal formulation Corollary 1.2 is stated as an 'immediate consequence' of Theorem 1.1; one line noting that d,α>1/3 implies α>f(d) (since f(d)<1/3 for d>1/3) would make this literally immediate.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: thresholds arise from matching explicit constructions to independent structural proofs, not from self-definition or fitted inputs.

full rationale

The paper is a self-contained extremal combinatorics argument. The functions f(d) and κ are obtained from explicit probabilistic constructions (Lemmas 3.2–3.3) that obstruct Hamiltonicity while preserving (n,d,μ)-density and the stated degree bounds; the matching sufficiency proofs (Theorems 1.1 and 1.4) proceed by a Hamilton-framework reduction to new dominant-component lemmas (4.1–4.3), which are proved from scratch via multicolour regularity, a weighted monochromatic-clique theorem (Lemma 5.3 / Theorem 5.4), and a codegree stability analysis (Theorem 5.8, Lemma 5.9). None of these steps defines the output in terms of itself, fits a parameter to data and re-labels it a prediction, or rests on a load-bearing uniqueness/ansatz result of the same authors. Citations to Han–Shu–Wang and Araújo–Piga–Schacht are to the open problems being resolved; the embedding black box (Lemma 2.2) is an external theorem of Lang–Sanhueza-Matamala. Standard matching of construction to proof is not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

The work sits on standard extremal-combinatorics foundations plus one recent external embedding theorem. No numerical parameters are fitted to data; the constants f(d) and κ arise from closed-form optimization of the constructions. Invented objects are definitional (framework families, weighted colorings) rather than physical entities.

assumptions (6)
  • domain assumption Lang–Sanhueza-Matamala Hamilton-framework embedding theorem (their Thm 3.5 specialized to k=t=3): local robust presence of a framework implies a tight Hamilton cycle.
    Invoked as Lemma 2.2; both main theorems reduce to it in §4.3. Not proved in this paper.
  • standard math Multicolor Szemerédi regularity lemma and standard weighted cut/triangle counting lemmas.
    Lemma 2.8–2.10; classical tools used throughout §§6–7.
  • standard math Lovász form of Kruskal–Katona (triangle bound) and Lenz–Mubayi perfect-matching theorem for uniformly dense 3-graphs with positive vertex degree.
    Lemmas 2.5 and 2.7; used for space condition and tail estimates.
  • standard math Uniform Turán density of C7^(3) is at most 4/27 (Bucić–Cooper–Kráľ–Mohr–Munhá Correia), so density >4/27 forces a 7-cycle.
    Lemma 2.6; supplies aperiodicity (F3) in the framework.
  • domain assumption Inheritance of (n,d,μ)-density and of minimum degree/codegree to almost all s-sets (Lang; Lang–Sanhueza-Matamala).
    Lemmas 2.3–2.4; quantitative bridge from global host to local frameworks.
  • domain assumption Definition of (n,d,μ)-density (one-sided box discrepancy / linear quasirandomness) as the ambient host class.
    Equation (1.1); standard weak quasirandomness notion from the literature the paper cites.
invented entities (3)
  • Weighted L-coloring of pairs and monochromatic triangle/cherry weights M, Ki
    purpose: Encode tight-component shadows after regularity so that a single dominant color can be selected.
    Definitional bookkeeping in §5; not an external physical entity. independent_evidence false because it is internal formalism.
  • Hamilton-framework family Ps with conditions (F1)–(F4)
    purpose: Local certificate that the embedding theorem can produce a tight Hamilton cycle.
    Specialized from Lang–Sanhueza-Matamala; used as the reduction target in §4. Internal to the method.
  • Dominant tight component C with quantitative shadow/density/degree lower bounds (Lemmas 4.1–4.2)
    purpose: Supply the spanning tightly connected piece that becomes F(H) inside almost every s-set.
    Main new structural objects; proved in §§6–7 from regularity plus weighted-clique/codegree stability. Falsifiable only inside the same combinatorial model.

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Pith. "Pith review of Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs." pith.science (2026). https://pith.science/paper/7K5JGZWJ

@misc{pith2026260723748,
  author       = {Pith},
  title        = {Pith review of: Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7K5JGZWJ}},
  note         = {Machine review of arXiv:2607.23748}
}
abstract

A $3$-uniform hypergraph (or $3$-graph) $H$ on $n$ vertices is \emph{$(n,d,\mu)$-dense} if $e_H(X,Y,Z)\ge d|X||Y||Z|-\mu n^3$ for all $X,Y,Z\subseteq V(H)$. This is one of the weakest standard notions of quasirandomness for $3$-graphs and is also known as linear quasirandomness. In this paper, we determine the sharp diagonal thresholds for tight Hamilton cycles in $(n,d,\mu)$-dense $3$-graphs $H$ under conditions on the minimum vertex degree $\delta_1(H)$ and the minimum codegree $\delta_2(H)$. We actually prove a general result: define \[ f(d):=\frac{1-\sqrt{(4d-1)/3}}2. \] We prove that $(n,d,\mu)$-density together with $\delta_1(H)\ge\alpha\binom{n-1}{2}$ forces a tight Hamilton cycle whenever $d > 1/3$ and $\alpha>f(d)$. In particular, $f(1/3)=1/3$, which answers Problem~8.3(i) of Ara\'ujo, Piga and Schacht and confirms Conjecture~8.1 of Han, Shu and Wang. For the minimum codegree condition, the sharp diagonal threshold is $(\kappa,\kappa)$, where $\kappa$ is the unique real solution of $\kappa=(1-\kappa)^3$. Since $\kappa\approx0.3177>1/4$, this gives a negative answer to Problem~8.3(ii) of Ara\'ujo, Piga and Schacht and disproves Conjecture~8.2 of Han, Shu and Wang. The two proofs use a common Hamilton-framework reduction, but the two degree conditions lead to distinct dominant-component lemmas for $(n, d, \mu)$-dense $3$-graphs, which are of independent interest and whose proofs do not rely on the absorption method.

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