Non-jumping densities of 3-uniform hypergraphs

Vaughn Komorech

REVIEW 4 major objections 4 minor 1 cited by

Non-jumping densities of 3-uniform hypergraphs

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proves that 64/81 and a parametric family of densities are non-jumps for 3-uniform hypergraphs.

desk verdict Promising new non-jump candidates via patterns, but a sign error in Theorem 4.8's c1 > 1/2 case leaves both main theorems unproven. read the letter →

arxiv 2511.07715 v2 pith:AHQXD3K5 submitted 2025-11-11 math.CO

Vaughn Komorech This is my paper

classification math.CO MSC 05C6505C35

keywords hypergraphlimitingdensityjumpnon-jumpLagrangianpatternblowupextremalcombinatorics

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 addresses the long-standing conjecture that every density in [0,1) is a jump for uniform hypergraphs — that is, that limiting edge densities cannot pile up arbitrarily close above any given density. A classical construction produced the first counterexamples, but the set of non-jumps remains poorly understood for r≥3. The paper develops a pattern-based sufficient condition for a density to be a non-jump for 3-uniform hypergraphs, following a recent refinement of the standard method. Using it, the author proves that 64/81 is not a jump, and that for every positive integer n the density 1 − (3n² − 2n + k³)/(n+k)³, with k=√(3n−2), is not a jump. If the proofs are correct, these are new additions to the short list of known non-jumps.

What carries the argument

The load-bearing object is the Lagrangian of an r-pattern, where a pattern is a hypergraph whose r-edges are multisets rather than sets. For a pattern P, its Lagrangian λ(P) is the maximum, over probability distributions on vertices, of the sum over edges of products of vertex weights (with multiplicities). The construction FR_v(P) takes a blowup of P in which one vertex is replaced by r copies and then adds a single edge on those copies; a known theorem says that if λ(FR_v(P)) equals λ(P) < 1 and v has positive weight in an optimal weighting of P, then r!·λ(P) is a non-jump. The paper's Theorem 4.8 identifies a family of 3-patterns for which this equality can be shown by a Lagrange-multipli

What would settle it

Numerically maximize the weight polynomial for FR_1(P) for the five-edge pattern P={123,122,112,113,223} over the simplex of vertex weights. If the maximum exceeds 32/243, then λ(FR_1(P)) > λ(P) and the proof of Theorem 1.5 fails; finding the maximum on the boundary with b=0 would support the paper's claim.

Watch

Extended reading notes

Core claim

The central claim is a template theorem: for any 3-pattern P that contains the edges {122} and {11i} for every other vertex i, if vertex 1 receives positive weight in an optimal weighting of P and the Lagrangian λ(P) is less than 1, then the density 3!·λ(P) is not a jump. The proof works by showing that the Lagrangian of the auxiliary object FR_1(P) constructed from P equals λ(P), which is precisely the sufficient condition for a non-jump. The author then applies the template to two patterns: a five-edge pattern with λ=32/243, giving the non-jump 64/81, and a family of patterns whose Lagrangians evaluate to the formula in Theorem 1.6. Thus the paper's discovery is a new method for producing

Load-bearing premise

The main theorem relies on the assertion that in any optimal weighting of the auxiliary object FR_1(P), the total weight on the original pattern's vertices is always greater than 1/2; the proof's handling of one case appears to contradict an earlier sign computation, so this load-bearing claim is not fully established.

Editorial extensions

If this is right

64/81 is now shown to be a non-jump, meaning there are families of 3-uniform hypergraphs with limiting densities strictly above 64/81 but arbitrarily close to it.
The family in Theorem 1.6 gives infinitely many non-jumps, with densities approaching 1 as n grows, so the set of non-jumps has a limit point at 1.
The template theorem means any 3-pattern containing the required edges and having Lagrangian below 1 automatically yields a non-jump, so the method converts a finite algebraic check into a new non-jump.
The exact Lagrangian computations pin down precise algebraic non-jumps rather than mere existence statements.

Reading between the lines

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

One could systematically enumerate 3-patterns satisfying the template and compute their Lagrangians numerically; this would likely reveal many more non-jumps and possibly a complete description of non-jumps near 1.
The template may be adaptable to r-uniform hypergraphs for r>3 by replacing the two distinguished edge types with their r-multiset analogues, though the proof as written is specific to r=3.
A direct numerical optimization of the Lagrangian of FR_1(P) for the five-edge pattern could independently confirm the equality λ(FR_1(P))=32/243; if it holds, the non-jump result is robust even if the proof's delicate step is later revised.
The infinite family suggests non-jumps accumulate at 1, raising the question of whether the set of non-jumps for 3-uniform hypergraphs is dense in some interval near 1.

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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

The paper extends Shaw's pattern-based approach to produce new non-jump densities for 3-graphs, and the headline examples (64/81 and an infinite family) are genuinely new if they hold. The pattern machinery is clean, the exposition is readable, and the Lagrangian computations for the examples check out. This is a real extension of the Frankl–Rödl/Shaw method, not a repackaging.

The problem is Theorem 4.8, the central lemma on which both Theorem 1.5 and Theorem 1.6 depend. The proof derives a derivative dk'/dc1 < 0 (positive numerator, negative denominator), then in the c1 > 1/2 case claims k' > 1/2 “since k increases when c1 increases.” That is exactly backwards: the derived sign says k' decreases as c1 increases, so for c1 > 1/2 you get k' < 1/2. That does not contradict a' > k', so the conclusion b' = 0 is not established. Without b' = 0, the lemma's claim λ(FR1(P)) = λ(P) does not follow, and Theorem 4.2 cannot be applied. The main results are unproven as written.

Smaller issues: Theorem 4.2 is cited from Shaw's unpublished work, so the external dependency should be flagged. Lemma 4.3 also has a slightly hand-wavy step about boundary cases. But these are minor next to the sign error.

If the sign error is fixable — perhaps the author intended a different argument for c1 > 1/2 — the approach could be worth reviving. As it stands, an expert referee would need to see that case redone. I would send it to review to let someone assess whether the proof can be patched, but I would not accept it in this form.

Referee Report

4 major / 4 minor

Summary. The paper develops a method for producing non-jump densities for 3-uniform hypergraphs using the Lagrangian of patterns and the Frankl–Rödl construction. The main results are Theorem 1.5, asserting that 64/81 is not a jump for r=3, and Theorem 1.6, asserting that for every n≥1, with k=√(3n−2), the density 1−(3n^2−2n+k^3)/(n+k)^3 is not a jump for r=3. The argument is organized around a key lemma, Theorem 4.8, which is supposed to show that for a 3-pattern P containing the edges {122} and {11i}, any optimal weighting of FR_1(P) satisfies b′=0, so that λ(FR_1(P))=λ(P). The paper then computes the Lagrangians of the specific patterns used in Theorems 1.5 and 1.6.

Significance. If the main theorems were established, they would provide new non-jump densities for 3-uniform hypergraphs and would illustrate a pattern-based method that could be useful for further constructions. The paper is clearly written and the Lagrangian computations for the examples are mostly explicit. However, the central lemma Theorem 4.8 has a serious gap in the c1>1/2 case, and both main theorems depend on that lemma. The paper also relies on Shaw's unpublished Theorem 4.2. In its present form, the correctness of the main results is not established, although the underlying approach remains promising.

major comments (4)

[§4, Theorem 4.8 (c1>1/2 case)] The proof derives dk′/dc1 = ((a′+b′)k′ + k′^2)/g with g<0, and the numerator is positive under the standing assumptions. Hence dk′/dc1<0: as c1 increases, the maximizing k′ decreases. The text then states the opposite: 'If c1 > 1/2, then we also get k′ > 1/2, since k increases when c1 increases'. This directly contradicts the derived sign. Therefore the conclusion k′>1/2 is not established in this case, and the contradiction a′>k′ is not obtained. Since b′=0 is forced only through that contradiction, Theorem 4.8 is unproved. Theorems 1.5 and 1.6 both invoke Theorem 4.8, so the main results are not supported as written.
[§4, Theorem 4.8 (monotonicity step)] Even apart from the sign error, the claim 'If c1 or c2 increases, then the k′ that maximizes w′(FR1(P)) must not decrease' is not derived. The implicit differentiation is performed only along the curve where c2 is at its lower bound, while at an actual optimum c2 is not constrained to equal that bound. A rigorous proof would need to control the coupled variation of c1 and c2, or derive the desired inequality directly from the Lagrange equations without this heuristic monotonicity step.
[§6, Theorem 1.6 (definition of P)] The definition of P as having edge set [n+1]^(3) ∪ {1,2,2} ∪ {1,3,3} ∪ ... ∪ {1,n+1,n+1} is inconsistent with the calculation that follows. The formula for w(P) contains the term w1^2 w2/2, which arises from edges of the form {1,1,i}. Those edges are not present in the stated edge set if [n+1]^(3) denotes ordinary 3-subsets. As written, the proof computes the Lagrangian of a different pattern from the one defined. This must be corrected for Theorem 1.6 to be proved.
[§3, Lemma 4.3] The proof claims that for equivalent vertices i and j, w(P) can be written as (wi+wj)C1 + (wi^2+wj^2)C2 + wiwj C3, with C1,C2,C3 depending on the other weights. This expression omits all monomials of total degree 3, such as wi^3, wj^3, wi^2 wj and wiwj^2, which can occur in a 3-pattern. The statement of the lemma may be salvageable by a symmetrization argument, but the proof as written is incomplete. Since Lemma 4.3 is used to justify the optimal-weighting reductions in Sections 4–6, the gap should be addressed.

minor comments (4)

[Abstract/Introduction] Typo: 'corallaries' should be 'corollaries'.
[§4 header] Typo: 'Frank-Röd l' should be 'Frankl–Rödl'.
[§6, Theorem 1.6] The notation [n+1]^(3) should be defined explicitly, since the proof depends on whether it means ordinary 3-subsets or something else.
[§4, Lemma 4.4] The equations labelled (4.5)–(4.7) would be clearer if they were consistently numbered and referenced; the current formatting is a minor readability issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's derivations are explicit Lagrangian computations; the only external dependency is Shaw's cited Theorem 4.2, which is an unverified assumption, not circular reasoning.

full rationale

I walked the derivation chain for Theorems 1.5 and 1.6. Each theorem reduces to a direct computation of lambda(P) for an explicit pattern, followed by an application of Theorem 4.2 and Theorem 4.8. The Lagrangian calculations in Sections 5 and 6 are self-contained constrained optimizations: no parameter is fitted to the target non-jump density, and no 'prediction' is extracted from the same data used to define the pattern. Lemma 4.4 proves the base case lambda(FR_1(P)) = 1/8 by a self-contained Lagrange multiplier argument, and Theorem 4.8 uses that as a base case, not as its conclusion. The only load-bearing external input is Shaw's Theorem 4.2, cited as '[10]' and not proved in this paper; that is a significant correctness/verification dependency, but it is not circular, since Shaw is not the present author and the cited result is not derived from the paper's own conclusions. I also note the apparent sign contradiction in the c1 > 1/2 branch of Theorem 4.8 that the reader flagged: the derived sign of dk'/dc1 seems to contradict the claim that k' > 1/2 in that branch. This is a mathematical gap or error, not circularity. There is no self-citation load-bearing step, no fitted input renamed as a prediction, no uniqueness theorem imported from the authors' prior work, and no known result merely renamed. Accordingly, the circularity score is 0.

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

The paper introduces no free parameters or invented entities; its load-bearing external input is the unpublished sufficient-condition theorem of Shaw.

assumptions (3)

domain assumption Theorem 4.2 (Frankl–Rödl sufficient condition) as stated in Shaw [10] is true.
The paper's method hinges on this external result, whose proof is not included and which is from an unpublished preprint.
standard math Turán densities exist and the Lagrangian formalism applies to patterns with the stated blowup limits.
Used throughout; standard in the area.
standard math Lemma 4.3's conclusion that equivalent vertices have equal weights at an optimum is valid.
Proved via Lagrange multipliers; appears correct.

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Cite this review

Pith. "Pith review of Non-jumping densities of 3-uniform hypergraphs." pith.science (2026). https://pith.science/paper/AHQXD3K5

@misc{pith2026251107715,
  author       = {Pith},
  title        = {Pith review of: Non-jumping densities of 3-uniform hypergraphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHQXD3K5}},
  note         = {Machine review of arXiv:2511.07715}
}

abstract

A density $\alpha\in [0, 1)$ is a jump for $r$ if there is some $c >0$ such that there does not exist a family of $r$-uniform hypergraphs with Tur\'an density in $(\alpha, \alpha + c)$. Erd\"os conjectured that all $\alpha\in [0, 1)$ are jumps for any $r$. This was disproven by Frankl and R\"odl when they provided examples of non-jumps. In this paper, we provide a method for finding non-jumps for $r = 3$ using patterns. As a direct consequence, we find a few more examples of non-jumps for $r = 3$.

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Forward citations

Cited by 1 Pith paper

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

The number $4/9$ is a non-jump for $3$-graphs
math.CO 2026-05 unverdicted novelty 7.0 of 10

4/9 is a non-jump for 3-graphs via a perturbed ABB construction inserting high-cogirth pairs of Steiner triple systems.

Reference graph

Works this paper leans on

11 extracted references · cited by 1 Pith paper

[1]

Hypergraphs do jump.Combinatorics, Probability and Com- puting, 20(2):161–171, 2011

Rahil Baber and John Talbot. Hypergraphs do jump.Combinatorics, Probability and Com- puting, 20(2):161–171, 2011

2011
[2]

P. Erd¨ os. On extremal problems of graphs and generalized graphs.Israel Journal of Mathe- matics, 2(3):183–190, 1964

1964
[3]

A note on the jumping constant conjecture of erd˝ os.Journal of Combinatorial Theory, Series B, 97(2):204–216, 2007

Peter Frankl, Yuejian Peng, Vojtech R¨ odl, and John Talbot. A note on the jumping constant conjecture of erd˝ os.Journal of Combinatorial Theory, Series B, 97(2):204–216, 2007

2007
[4]

Hypergraphs do not jump.Combinatorica, 4(2):149–159, 1984

Peter Frankl and Vojtˇ ech R¨ odl. Hypergraphs do not jump.Combinatorica, 4(2):149–159, 1984

1984
[5]

Generating non-jumps from a known one.Science China Mathematics, 67(12):2899–2908, 2024

Jianfeng Hou, Heng Li, Caihong Yang, and Yixiao Zhang. Generating non-jumps from a known one.Science China Mathematics, 67(12):2899–2908, 2024

2024
[6]

London Mathematical Society Lec- ture Note Series

Peter Keevash.Hypergraph Tur´ an problems, page 83–140. London Mathematical Society Lec- ture Note Series. Cambridge University Press, 2011. 10 V AUGHN KOMORECH

2011
[7]

Co-degree density of hypergraphs.Journal of Combinatorial Theory, Series A, 114(6):1118–1132, 2007

Dhruv Mubayi and Yi Zhao. Co-degree density of hypergraphs.Journal of Combinatorial Theory, Series A, 114(6):1118–1132, 2007

2007
[8]

Using lagrangians of hypergraphs to find non-jumping numbers (i).Annals of Combinatorics, 12(3):307–324, 2008

Yuejian Peng. Using lagrangians of hypergraphs to find non-jumping numbers (i).Annals of Combinatorics, 12(3):307–324, 2008

2008

Show all 11 references

[9]

On jumping densities of hypergraphs.Graphs and Combinatorics, 25(5):759– 766, 2009

Yuejian Peng. On jumping densities of hypergraphs.Graphs and Combinatorics, 25(5):759– 766, 2009

2009
[10]

Minimal hypergraph non-jumps, 2025

Benedict Randall Shaw. Minimal hypergraph non-jumps, 2025

2025
[11]

Non-jumping tur´ an densities of hypergraphs.Discrete Mathe- matics, 346(1):113195, 2023

Zilong Yan and Yuejian Peng. Non-jumping tur´ an densities of hypergraphs.Discrete Mathe- matics, 346(1):113195, 2023

2023

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Hypergraphs do jump.Combinatorics, Probability and Com- puting, 20(2):161–171, 2011

Rahil Baber and John Talbot. Hypergraphs do jump.Combinatorics, Probability and Com- puting, 20(2):161–171, 2011

2011

[2] [2]

P. Erd¨ os. On extremal problems of graphs and generalized graphs.Israel Journal of Mathe- matics, 2(3):183–190, 1964

1964

[3] [3]

A note on the jumping constant conjecture of erd˝ os.Journal of Combinatorial Theory, Series B, 97(2):204–216, 2007

Peter Frankl, Yuejian Peng, Vojtech R¨ odl, and John Talbot. A note on the jumping constant conjecture of erd˝ os.Journal of Combinatorial Theory, Series B, 97(2):204–216, 2007

2007

[4] [4]

Hypergraphs do not jump.Combinatorica, 4(2):149–159, 1984

Peter Frankl and Vojtˇ ech R¨ odl. Hypergraphs do not jump.Combinatorica, 4(2):149–159, 1984

1984

[5] [5]

Generating non-jumps from a known one.Science China Mathematics, 67(12):2899–2908, 2024

Jianfeng Hou, Heng Li, Caihong Yang, and Yixiao Zhang. Generating non-jumps from a known one.Science China Mathematics, 67(12):2899–2908, 2024

2024

[6] [6]

London Mathematical Society Lec- ture Note Series

Peter Keevash.Hypergraph Tur´ an problems, page 83–140. London Mathematical Society Lec- ture Note Series. Cambridge University Press, 2011. 10 V AUGHN KOMORECH

2011

[7] [7]

Co-degree density of hypergraphs.Journal of Combinatorial Theory, Series A, 114(6):1118–1132, 2007

Dhruv Mubayi and Yi Zhao. Co-degree density of hypergraphs.Journal of Combinatorial Theory, Series A, 114(6):1118–1132, 2007

2007

[8] [8]

Using lagrangians of hypergraphs to find non-jumping numbers (i).Annals of Combinatorics, 12(3):307–324, 2008

Yuejian Peng. Using lagrangians of hypergraphs to find non-jumping numbers (i).Annals of Combinatorics, 12(3):307–324, 2008

2008