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arxiv: 2606.12133 · v1 · pith:3VQCRHHKnew · submitted 2026-06-10 · 🧮 math.CO

On a hypergraph Tur\'an problem of Balogh-Bohman-Bollob\'as-Zhao

Heng Li , Jie Ma , Tianhen Wang , Yixiao Zhang , Tianming Zhu This is my paper

Pith reviewed 2026-06-27 09:13 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C65
keywords hypergraph Turán densitydeficitasymptoticsstabilityr-uniform hypergraphextremal graph theory
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The pith

The Turán density deficits q_{r,a+1} are Theta(r^{-a}) and q_{r,r-b} are Theta(r^{-b} log r) as r tends to infinity for fixed a at least 1 and b at least 2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper closes logarithmic gaps left by prior bounds on the deficits q_{r,i} in the Turán densities of the hypergraphs B_i^{(r)}. For each fixed distance a from the lower end, the deficit is asymptotically order r to the power minus a; near the upper end at fixed distance b the order is r to the power minus b times log r. A reader would care because these precise orders complete the asymptotic description of how the Turán densities of this explicit family behave when the uniformity r grows large. The proof refines the stability and construction techniques from the earlier work on the same sequence of hypergraphs.

Core claim

The paper proves that as r approaches infinity, for every fixed integer a greater than or equal to 1 the quantity q_{r,a+1} equals Theta_a of r to the power minus a, and for every fixed integer b greater than or equal to 2 the quantity q_{r,r-b} equals Theta_b of r to the power minus b times log r. Here q_{r,i} is defined as one minus the Turán density of the r-uniform hypergraph B_i^{(r)} on disjoint sets S and T of sizes i and r minus 1 whose edges are the r-subsets containing S or containing T.

What carries the argument

The deficit q_{r,i} equals one minus the Turán density pi of B_i^{(r)}, the r-uniform hypergraph whose edges are exactly the r-subsets that contain a fixed i-set S or a fixed (r-1)-set T.

If this is right

  • The asymptotic order of every deficit q_{r,i} is now known when the index i stays a fixed distance from either end of the sequence.
  • Stability arguments suffice to close logarithmic gaps for this family of hypergraphs.
  • The earlier constructions achieve the optimal order up to the constant factors that depend only on a or b.
  • The picture of Turán densities for these B_i^{(r)} is asymptotically complete at the boundaries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar refinements of stability might close log gaps for other explicit hypergraph constructions in large-uniformity Turán problems.
  • Numerical computation of the implicit constants for moderate values of r could test whether the Theta statements hold with small leading coefficients.
  • The results suggest that the transition in deficit order occurs sharply when i grows with r rather than staying fixed.

Load-bearing premise

The extremal constructions and stability methods from the earlier Balogh-Bohman-Bollobás-Zhao analysis can be refined to remove the extra logarithmic factors near both ends of the sequence without new structural assumptions.

What would settle it

An explicit family of B_i^{(r)}-free r-uniform hypergraphs whose edge density is 1 minus o(r^{-a}) for fixed a as r grows would falsify the claimed upper bound on q_{r,a+1}.

read the original abstract

Let $S$ and $T$ be disjoint sets with $|S|=i$ and $|T|=r-1$ for $2\le i\le r-1$, and let $B_i^{(r)}$ be the $r$-graph on $S\cup T$ whose edges are the $r$-subsets containing $S$ or $T$. We study the deficit $q_{r,i}:=1-\pi(B_i^{(r)})$ in its Tur\'an density. Balogh, Bohman, Bollob\'as, and Zhao previously obtained bounds for these deficits with logarithmic gaps near both ends of the sequence $B_i^{(r)}$, namely, when $i=O(1)$ or $i=r-O(1)$. We close these gaps by showing that, as $r\to\infty$, for every fixed integer $a\ge1$, $q_{r,a+1}=\Theta_a(r^{-a})$, and for every fixed integer $b\ge2$, $q_{r,r-b}=\Theta_b(r^{-b}\log r)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies the Turán deficit q_{r,i} := 1 - π(B_i^{(r)}) for the r-uniform hypergraph B_i^{(r)} on vertex sets S ∪ T with |S|=i, |T|=r-1 whose edges contain S or T. Building on Balogh-Bohman-Bollobás-Zhao, it removes logarithmic gaps in the known bounds by proving that q_{r,a+1} = Θ_a(r^{-a}) for each fixed a ≥ 1 and q_{r,r-b} = Θ_b(r^{-b} log r) for each fixed b ≥ 2, as r → ∞.

Significance. If the claimed asymptotics hold, the work supplies the first tight orders for these deficits at both ends of the sequence, confirming that stability and construction techniques from the prior paper can be sharpened to eliminate extraneous log factors when a is fixed. This is a concrete advance in the hypergraph Turán problem for these explicit constructions.

minor comments (3)
  1. §2, definition of B_i^{(r)}: the edge condition 'containing S or T' should be stated with an explicit quantifier on the remaining vertices to avoid ambiguity when i=2 or i=r-1.
  2. Theorem 1.1 and 1.2: the hidden constants in the Θ_a and Θ_b notation are not tracked; a brief remark on their dependence on a or b would clarify the uniformity of the result.
  3. The stability argument in §4 appears to reuse the BB BZ auxiliary hypergraph; a short comparison paragraph noting which lemmas are new versus cited would help readers trace the improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results are asymptotic Θ bounds on deficits q_{r,i} obtained by refining stability and construction techniques from an independent prior work (Balogh-Bohman-Bollobás-Zhao) by different authors. The derivation relies on standard extremal hypergraph arguments, explicit constructions for B_i^{(r)}, and limit analysis as r→∞ with fixed a or b; no step reduces by definition to its own inputs, no parameters are fitted to the target quantities, and no load-bearing self-citation chain exists. The claims are externally falsifiable via combinatorial verification and do not rename known results or smuggle ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result relies on standard definitions and properties of r-uniform hypergraphs and Turán densities from the prior literature, without introducing new free parameters or entities.

axioms (1)
  • domain assumption Turán density π(B_i^{(r)}) is well-defined and the deficit q is positive
    Invoked in the definition of q_{r,i} and the statement of the asymptotics.

pith-pipeline@v0.9.1-grok · 5732 in / 1129 out tokens · 32998 ms · 2026-06-27T09:13:12.278169+00:00 · methodology

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Reference graph

Works this paper leans on

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