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Intervals of hypergraph Tur\'an densities

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Pith's one-line read The set of Turán densities for r-graphs includes non-degenerate intervals for every r at least 3.

desk verdict This paper shows that Turán density sets for r-graphs contain intervals of positive length by explicit construction, resolving the 2007 and 2016 questions. read the letter →

arxiv 2605.25914 v1 pith:SXJNYMRZ submitted 2026-05-25 math.CO

Xizhi Liu , Oleg Pikhurko This is my paper
classification math.CO
keywords Turándensityhypergraphr-graphinfinitefamilyHausdorffdimensionextremalsettheoryinterval
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 establishes that the collection of all possible Turán densities for r-uniform hypergraphs, when infinite families are permitted, contains intervals of positive length. In particular there is an interval of densities running from some value less than 1 up to 1 itself. A reader would care because this reveals that these densities are not just discrete points but can vary continuously in ranges, which was previously unknown. It answers questions posed in 2007 and 2016 and shows the Hausdorff dimension reaches its maximum value of 1. The result also implies that densities achievable by finite families alone are dense in an interval for 3-graphs.

What carries the argument

Constructions of possibly infinite families of r-graphs whose extremal densities can be varied continuously to fill intervals in Π^(r)_∞.

What would settle it

An explicit r and a positive-length gap in achievable densities near 1 that no family of r-graphs can fill.

Watch

Extended reading notes

Core claim

For every integer r ≥ 3, the set Π^(r)_∞ of Turán densities of (possibly infinite) families of r-graphs contains non-degenerate intervals, including an interval of the form [1-δ_r,1] for some δ_r>0. This shows that the Hausdorff dimension of Π^(r)_∞ is 1, and the set of uniform Turán densities of finite families of 3-graphs is dense in a non-degenerate interval.

Load-bearing premise

The standard limit-superior definition of Turán density extends in a way that permits continuous variation when families are allowed to be infinite.

Editorial extensions

If this is right

  • The Hausdorff dimension of Π^(r)_∞ equals 1 for every r ≥ 3.
  • Uniform Turán densities of finite 3-graph families are dense in a non-degenerate interval.
  • Questions of Frankl-Peng-Rödl-Talbot (2007) and Grosu (2016) on the structure of these sets receive affirmative answers.

Reading between the lines

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

  • The same interval phenomenon may hold for other notions of hypergraph extremal density that allow infinite forbidden families.
  • Approximation algorithms for Turán numbers in large r-graphs could exploit the density intervals to guarantee solutions in certain ranges.
  • Whether finite families alone already produce intervals for r > 3 remains open and could be tested by searching for gaps in small cases.
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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

0 major / 2 minor

Summary. The paper proves that for every integer r≥3, the set Π^(r)_∞ of Turán densities of (possibly infinite) families of r-graphs contains non-degenerate intervals, including an interval of the form [1-δ_r,1] for some δ_r>0. This answers a question of Frankl, Peng, Rödl and Talbot from 2007. It also shows that the Hausdorff dimension of Π^(r)_∞ is 1, resolving a question of Grosu from 2016. Additionally, the set of uniform Turán densities of finite families of 3-graphs is dense in a non-degenerate interval.

Significance. If the results hold, this work substantially clarifies the structure of the set of hypergraph Turán densities by demonstrating that it contains intervals and achieves the maximum possible Hausdorff dimension of 1. The explicit constructions using one-parameter families of forbidden r-graphs, combined with supersaturation and stability arguments reducing to finite subgraphs and tunable constraints, provide a robust foundation. The use of standard limit-superior definitions without additional conjectures strengthens the contribution.

minor comments (2)
  1. [§1] The notation for the limit superior in the definition of π(F) for infinite F could be stated explicitly in §1 to make the extension from the finite case immediate.
  2. [§4] In the continuity argument for π(F_α), the dependence on the stability theorem is invoked without a self-contained reference to the precise version used; adding a citation or brief recap would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, detailed summary of the results, and recommendation to accept.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; existence via explicit construction

full rationale

The paper establishes that Π^(r)_∞ contains intervals by exhibiting explicit one-parameter families F_α of (possibly infinite) r-graphs and proving that their Turán densities π(F_α) form a continuous monotone interval via the standard lim sup definition of ex(n,F)/binom(n,r) together with supersaturation and stability reductions to finitely many forbidden subgraphs. No step equates a derived quantity to a fitted parameter, renames a known result, or relies on a self-citation whose content is itself the target claim. The cited prior questions (Frankl-Peng-Rödl-Talbot 2007, Grosu 2016) are external and the proofs are self-contained against the usual extremal definitions.

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

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; the result is framed as a pure existence proof within standard extremal combinatorics.

assumptions (1)
  • standard math Standard axioms of ZFC set theory and the usual definitions of r-graphs and Turán density as lim sup of maximum densities
    The statement presupposes that the Turán density is well-defined for families of r-graphs.

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Cite this review

Pith. "Pith review of Intervals of hypergraph Tur\'an densities." pith.science (2026). https://pith.science/paper/SXJNYMRZ

@misc{pith2026260525914,
  author       = {Pith},
  title        = {Pith review of: Intervals of hypergraph Tur\'an densities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXJNYMRZ}},
  note         = {Machine review of arXiv:2605.25914}
}
abstract

We prove that, for every integer $r\ge 3$, the set $\Pi^{(r)}_\infty$ of Tur\'an densities of (possibly infinite) families of $r$-graphs contains non-degenerate intervals, including an interval of the form $[1-\delta_r,1]$ for some $\delta_{r}>0$. This answers a question of Frankl, Peng, R\"odl and Talbot from 2007. This also shows that the Hausdorff dimension of $\Pi^{(r)}_\infty$ has the maximum possible value 1, thus resolving a question of Grosu from 2016, whereas previously it was not even known whether it is non-zero. We also derive that the set of uniform Tur\'an densities of finite families of $3$-graphs is dense in a non-degenerate interval.

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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. Tree suspensions and transfer functions for single degree Tur\'an spectra

    math.CO 2026-07 accept novelty 7.5 of 10

    Tree suspensions realize three transfer functions on single degree Turán spectra, yielding infinitely many accumulation points for all parameters and arbitrarily high algebraic degree for ordinary and half-or-higher d...

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