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The number $4/9$ is a non-jump for $3$-graphs

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Pith's one-line read The number 4/9 is a non-jump for 3-uniform hypergraphs.

desk verdict Liu and Mubayi prove 4/9 is a non-jump for 3-graphs by inserting a high-cogirth Steiner triple system pair into the ABB B-part. read the letter →

arxiv 2605.13567 v1 pith:B546OLTU submitted 2026-05-13 math.CO

Xizhi Liu , Dhruv Mubayi This is my paper
classification math.CO
keywords non-jump3-uniformhypergraphsSteinertriplesystemsABBpatternFrankl-RodlmethodErdosjumpproblemTurándensities
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 proves that 4/9 is a non-jump for 3-uniform hypergraphs. This means there exist 3-graphs with edge densities arbitrarily close to 4/9 that avoid containing any 3-graph of strictly higher density. The proof uses a construction that perturbs the ABB pattern by inserting the union of a high-cogirth pair of Steiner triple systems into the B-part. This surpasses the previous barrier obtained from finite-pattern versions of the Frankl-Rödl method. The authors conjecture that 4/9 is the smallest non-jump for 3-graphs, which would resolve an old question of Erdős in strong form.

What carries the argument

The ABB pattern perturbed by inserting the union of a high-cogirth pair of Steiner triple systems into the B-part.

What would settle it

A direct computation showing that the perturbed construction forces a subhypergraph whose density exceeds 4/9 would disprove the non-jump claim.

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Extended reading notes

Core claim

We prove that 4/9 is a non-jump for 3-uniform hypergraphs. Our construction perturbs the ABB pattern by inserting, inside the B-part, the union of a high-cogirth pair of Steiner triple systems. This goes below the barrier for non-jumps obtainable by Shaw's finite-pattern formulation of the Frankl--Rödl method introduced in 1984. All results employing this approach use patterns where one of the parts has complete shadow. As the ABB pattern is the smallest one with this property, the value 4/9 is the natural barrier using this technique, and we conjecture that 4/9 is the smallest non-jump for 3-graphs.

Load-bearing premise

Inserting the union of a high-cogirth pair of Steiner triple systems into the B-part of the ABB pattern produces a valid construction that achieves the non-jump property at 4/9.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper claims to prove that 4/9 is a non-jump for 3-uniform hypergraphs. The proof proceeds by perturbing the ABB pattern: the B-part is replaced by the union of a high-cogirth pair of Steiner triple systems, producing a sequence of 3-graphs whose edge densities approach 4/9 from below while preserving the non-jump property. This construction is asserted to surpass the barrier obtainable from Shaw's finite-pattern version of the Frankl-Rödl method.

Significance. If the central construction is valid, the result is significant: it supplies the smallest non-jump value achievable by the complete-shadow pattern technique and conjectures that 4/9 is the minimal non-jump for 3-graphs, thereby answering Erdős's question in strong form. The argument is direct and combinatorial, relying on external properties of Steiner triple systems rather than fitted parameters or self-referential definitions.

major comments (2)
  1. [§3] The density calculation after inserting the high-cogirth STS pair into the B-part of the ABB pattern (main construction, §3) is not carried out explicitly; it is asserted that the limit superior equals 4/9 from below, but the precise contribution of the inserted edges to the overall density and the control of shadows must be verified step-by-step to confirm the approach does not overshoot.
  2. [Proof of Theorem 1.1] The claim that the perturbed pattern preserves the non-jump property (proof of Theorem 1.1) rests on the high-cogirth condition preventing new 3-edges that would force the Turán density above 4/9; the intersection and shadow arguments are sketched but lack the concrete forbidden-subgraph checks needed to establish that the construction remains valid at the limit.
minor comments (1)
  1. [Introduction] The statement that the ABB pattern is the smallest with complete shadow would benefit from a brief reference to the relevant prior result or a short justification in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address each major comment below and will incorporate the requested clarifications into the revised version.

read point-by-point responses

  1. Referee: [§3] The density calculation after inserting the high-cogirth STS pair into the B-part of the ABB pattern (main construction, §3) is not carried out explicitly; it is asserted that the limit superior equals 4/9 from below, but the precise contribution of the inserted edges to the overall density and the control of shadows must be verified step-by-step to confirm the approach does not overshoot.

    Authors: We agree that an explicit step-by-step density calculation is needed for clarity. In the revised manuscript we will add a detailed computation showing the precise edge contribution of the high-cogirth STS pair within the B-part, combined with the ABB edges, and verify that the resulting density sequence approaches 4/9 strictly from below while the shadows remain controlled. revision: yes

  2. Referee: [Proof of Theorem 1.1] The claim that the perturbed pattern preserves the non-jump property (proof of Theorem 1.1) rests on the high-cogirth condition preventing new 3-edges that would force the Turán density above 4/9; the intersection and shadow arguments are sketched but lack the concrete forbidden-subgraph checks needed to establish that the construction remains valid at the limit.

    Authors: We accept that the proof sketch would be strengthened by explicit checks. We will expand the argument in the proof of Theorem 1.1 to include concrete forbidden-subgraph verifications, spelling out the intersection and shadow calculations that show how the high-cogirth property prevents any new 3-edges from forcing the Turán density above 4/9. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct construction from external STS properties

full rationale

The derivation consists of an explicit combinatorial construction: perturbing the ABB pattern by inserting a high-cogirth pair of Steiner triple systems into the B-part to produce 3-graphs whose densities approach 4/9 from below while preserving the non-jump property. This relies on known external properties of Steiner triple systems (cogirth, intersections, shadows) rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The ABB pattern and Frankl-Rödl/Shaw framework are cited as prior context, but the central claim is the new insertion step, which is presented as a self-contained argument against external combinatorial benchmarks. No step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claim rests on standard properties of Steiner triple systems and the ABB pattern from prior literature; no free parameters or new entities are introduced.

assumptions (1)
  • domain assumption Existence and insertability of high-cogirth pairs of Steiner triple systems into the ABB pattern
    Invoked to achieve densities approaching 4/9 without creating a jump.

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Pith. "Pith review of The number $4/9$ is a non-jump for $3$-graphs." pith.science (2026). https://pith.science/paper/B546OLTU

@misc{pith2026260513567,
  author       = {Pith},
  title        = {Pith review of: The number $4/9$ is a non-jump for $3$-graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B546OLTU}},
  note         = {Machine review of arXiv:2605.13567}
}
abstract

We prove that $4/9$ is a non-jump for $3$-uniform hypergraphs. Our construction perturbs the $ABB$ pattern by inserting, inside the $B$-part, the union of a high-cogirth pair of Steiner triple systems. This goes below the barrier for non-jumps obtainable by Shaw's finite-pattern formulation of the Frankl--R\"odl method introduced in 1984. All results employing this approach use patterns where one of the parts has complete shadow. As the $ABB$ pattern is the smallest one with this property, the value $4/9$ is the natural barrier using this technique, and we conjecture that $4/9$ is the smallest non-jump for $3$-graphs. If our conjecture is true, this would answer (in a very strong form) an old question of Erd\Hos.

Figures

Figures reproduced from arXiv: 2605.13567 by the authors.

Figure 1
Figure 1. Current partial picture for jumps and non-jumps of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Current partial picture for jumps and non-jumps of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

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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.

  1. Intervals of hypergraph Tur\'an densities

    math.CO 2026-05 unverdicted novelty 8.0 of 10

    The set of Turán densities of (possibly infinite) families of r-graphs contains non-degenerate intervals for every r≥3.

Reference graph

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    Let f(b) := 3cb2(1−b) + 6qb 3

    Henceq≤c/6. Let f(b) := 3cb2(1−b) + 6qb 3. Then f ′(b) = 3b 2c−3(c−2q)b . Since q≤c/ 6, the critical pointb0 = 2c 3(c−2q) lies in[0 , 1]. Thus the maximum off on[0 , 1]is attained at b0, and a direct substitution gives f(b 0) = 4c3 9(c−2q) 2 . Thereforef(b 0)≤4/9is equivalent ...

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