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The largest Fano-plane-free piece of a random 3-uniform hypergraph is bipartite exactly above an explicit density threshold.

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The largest Fano-free subhypergraph of G_{n,p}^{(3)} is bipartite whp precisely above the sharp threshold p̂ = Θ_F n^{-2/3}(log n)^{1/6}.

T0 review reviewed 2026-07-31 challenge →

load-bearing objection First genuine sharp Turán threshold in random hypergraphs; the Fano case is fully worked and the architecture looks transferable.

arxiv 2607.28071 v1 pith:QSHCHK4Q submitted 2026-07-30 math.CO math.PR

Random Tur\'an Theorem for the Fano Plane

Ilay Hoshen This is my paper
classification math.CO math.PR MSC 05C6505C8005D05
keywords Fano planerandom hypergraphsTurán theoremsharp thresholdstabilitycores3-uniform hypergraphs
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Classical extremal hypergraph theory says the biggest Fano-plane-free 3-uniform hypergraph on n vertices is bipartite. This paper asks the same question inside the random hypergraph G_{n,p}^{(3)}. It pins down an explicit constant Θ_F and the precise window ˆp = Θ_F n^{-2/3} (log n)^{1/6} at which the property flips: above (1+ε)ˆp every largest Fano-free subhypergraph is bipartite with high probability, while below (1-ε)ˆp it is not. The result is the first sharp threshold of its kind for a Turán-type problem in random hypergraphs. A sympathetic reader cares because it shows that the deterministic structural theorem survives random noise down to a clean, computable density, and that the same stability-plus-rigidity toolkit that worked for graphs now works for at least one nontrivial hypergraph.

Core claim

There exists an explicit constant Θ_F, defined from the asymptotic number of Fano copies in a nearly balanced complete bipartite 3-graph plus one edge, such that the property “every largest F-free subhypergraph of G_{n,p}^{(3)} is bipartite” holds with high probability precisely when p exceeds (1+ε)Θ_F n^{-2/3}(log n)^{1/6} and fails when p is smaller than (1-ε) times that quantity (down to 1/n²).

What carries the argument

The (C,d,α)-core of a hypergraph: two large vertex sets that sit inside opposite sides of every cut of deficit at most d. Combined with a stability theorem that every largest F-free piece is already nearly bipartite, the core reduces the problem to a matching argument that forbids adding any internal edge without creating an Fano plane.

Load-bearing premise

The argument for the upper side of the threshold begins from a stability result that already assumes every largest Fano-free piece is within o(total edges) of bipartite once p is a constant times n^{-2/3}; if that near-bipartiteness fails at the precise logarithmic scale, the rest of the reduction collapses.

What would settle it

Compute, for a sequence of n and p just above and just below the predicted ˆp, the size of a maximum bipartite subhypergraph versus the size of a maximum Fano-free subhypergraph obtained by a simple local-search or integer-program heuristic; if the two sizes coincide above ˆp and diverge below it for large n, the threshold is confirmed.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The same core-and-stability method is expected to yield sharp thresholds for any hypergraph whose deterministic Turán problem is already solved and whose extremal examples are multipartite.
  • The explicit constant Θ_F can be evaluated numerically from the limit density of Fano copies in K_2^+(m), giving a concrete numerical prediction for simulations.
  • Below the threshold one can always enlarge a maximum cut by a single internal edge without creating an Fano plane, so the extremal function is strictly larger than the bipartite Turán number.
  • The 0-statement holds already for p as small as n^{-2+δ}, showing the bipartite structure is forced only near the appearance of dense Fano copies.

Where Pith is reading between the lines

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

  • The appearance of the first sharp hypergraph threshold suggests that once stability is known, the graph-theoretic switching and rigidity arguments transfer with only routine changes in density exponents.
  • If an analogous stability theorem is proved for the generalised triangle F_5, the same proof outline would immediately give a sharp threshold for 3-partiteness in random 3-graphs.
  • The logarithmic power 1/6 is exactly 1/(e(F)-1), the same universal exponent that appears for graphs; this hints that the exponent is determined solely by the balanced density and not by higher uniformity.
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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 / 6 minor

Summary. The paper establishes a sharp threshold for the property that every largest Fano-plane-free subhypergraph of the binomial random 3-uniform hypergraph G_{n,p}^{(3)} is bipartite. Writing F for the Fano plane and defining an explicit constant Θ_F via the limit π_F = lim N(F, K_2^+(m))/m^{v(F)-3} and the algebraic relation (2), the authors prove that for p ≥ (1+ε)Θ_F n^{-2/3}(log n)^{1/6} (with p = o(1)) the property holds whp, while for 1/n^{2} ≪ p ≤ (1-ε)Θ_F n^{-2/3}(log n)^{1/6} it fails whp. The 1-statement proceeds from Conlon–Gowers stability, extraction of low-/high-degree coloured subgraphs Q, fixed-cut Janson bounds, a rigidity/core argument, and a switching lemma that absorbs the union bound over cuts; the 0-statement uses core existence, resampling of F\{e}-copies, and a second-moment argument on internal edges free of such copies.

Significance. This is, to the best of current knowledge, the first sharp-threshold result for a Turán-type problem in random hypergraphs. The deterministic extremal structure for the Fano plane (bipartite) has been known since Frankl–Füredi / Keevash–Sudakov, but transferring the sharp random-graph architecture of DeMarco–Kahn, Hoshen–Samotij and Hoshen–Samotij–Zhukovskii to the 3-uniform setting requires substantial new work: hypergraph-adjusted rigidity (Theorem 6.4 / Corollary 6.5), low- and high-degree Janson estimates (Lemmas 5.1–5.2), a sparsification lemma for the intermediate-density regime (Lemma 7.3), and a full switching argument (Lemma 7.2 / §8). The constant Θ_F is combinatorial rather than fitted, and the proof is written out in full. The result is a clear advance and supplies a template that the authors reasonably expect to extend to other hypergraphs whose deterministic extremal theory is settled.

minor comments (6)
  1. [§2.1 and after (6)] The constant hierarchy in §2.1 is helpful but the writing of β = β(n) ≪ 1 (invoked after (6) and in the definition of d_Q) is slightly loose: Conlon–Gowers supplies a fixed β > 0 for any p ≥ C n^{-2/3}, and a fixed small β already gives the room needed in Claim 2.4. Clarifying that a fixed β suffices would remove a minor source of confusion.
  2. [throughout] Numerous typographical and grammatical slips accumulate in a long technical manuscript (e.g., “challanging”, “crictical”, “guarenteed”, “equiqqed”, “faimly”, “cemtral”, “corrolary”, “estemate”, “accordinly”). A careful copy-edit pass is needed before publication.
  3. [§1, (1)] In the definition of π_F (display (1)) the host is written K_2^+(m) while the surrounding text speaks of parts of size a; the notation should be made consistent (parts of size m).
  4. [§2, Claim 2.2] Claim 2.2 item (3) and Figure 2 describe the star configuration; a one-sentence reminder that the same centres also send stars into A_2 would make the subsequent high-degree Janson analysis (Lemma 5.2) easier to follow on a first reading.
  5. [§7.2–7.3] Table 1 (parameter summary) is useful; adding a short pointer to it at the beginning of §7.3 would help the reader navigate the case distinctions when applying Lemma 7.2.
  6. [§1] Several references to “[11, 12]” and “[7]” are used for technique transfer; a single sentence in the introduction explicitly listing which lemmas are new versus which are hypergraph adaptations would improve transparency for non-specialists.

Circularity Check

0 steps flagged

No significant circularity: Θ_F is a purely combinatorial constant and the sharp-threshold argument is a self-contained hypergraph adaptation of known graph techniques plus an external stability black-box.

full rationale

The constant Θ_F is defined from the combinatorial limit π_F = lim N(F, K_2^+(m))/m^{v(F)-3} and the algebraic relation (2); it is not fitted to any data or simulation. The 1-statement opens from the external Conlon–Gowers stability theorem (Theorem 1.4 / [5]), extracts low/high-degree Q-subgraphs, applies Janson estimates on fixed cuts (Lemmas 5.1–5.2), and closes via a rigidity-switching argument written out in full for hypergraphs (Sections 6–8). The 0-statement uses core existence (Corollary 6.5) and a second-moment argument on F\{e} copies, again written out. Self-citations to the author’s graph papers [11,12] supply proof architecture and switching lemmas, not a uniqueness theorem or numerical threshold that forces the Fano result by definition. No equation equates a claimed prediction to a fitted input, and no load-bearing step reduces to its own conclusion. Ordinary long-proof risk remains, but that is not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 3 invented entities

The result rests on standard probabilistic tools, the classical deterministic Fano Turán theorem, Conlon–Gowers sparse stability, and the rigidity/core technology developed for graphs. No free parameters are fitted to data; all constants are either combinatorial (π_F, m_3(F)=3/2) or existentially chosen small/large in a declared hierarchy. Technical devices (Q-families, (C,d)-cores, sparsified F_q) are proof artefacts, not ontological inventions.

axioms (6)
  • domain assumption Frankl–Füredi / Keevash–Sudakov: the largest Fano-free subhypergraph of K_n^{(3)} is bipartite.
    Deterministic base case; invoked throughout the introduction and to justify that bipartite subgraphs are admissible F-free objects.
  • domain assumption Conlon–Gowers stability (Thm 1.4): for p ⩾ C n^{-2/3}, every largest F-free subhypergraph of G_{n,p}^{(3)} is within β n³ p edges of bipartite.
    Starting point of the entire 1-statement reduction in §2; without the o(n³ p) internal-edge bound the Q-extraction and deficit comparison fail.
  • standard math Janson’s inequality and its matching-number corollaries (Thm 3.1, Cor 3.2–3.3).
    Primary tool for lower-bounding the number of edge-disjoint Q-supported crossing Fano copies once a cut is fixed.
  • domain assumption Fano plane is strictly 3-balanced with m_3(F)=3/2 and is not bipartite (linear, every pair in one edge).
    Used in Claim 4.4 and all Δ_p estimates to obtain the extra n^λ factors that keep the second-moment ratio small.
  • standard math Harris inequality / FKG for increasing and decreasing events on the same edge set (Lemma 6.3).
    Correlates the existence of a (C,d)-core with monotone events determined by the external edges of the core.
  • domain assumption Hypergraph rigidity and core existence for G_{n,m}^{(3)} (Thm 6.4 / Cor 6.5), adapted from DeMarco–Kahn and Hoshen–Samotij–Zhukovskii.
    Supplies the unique large core that lets the union bound over cuts be replaced by a bound over Q only; proof is given but relies on the graph-case template.
invented entities (3)
  • (C,d,α)-rigidity and (C,d)-core of a 3-uniform hypergraph independent evidence
    purpose: Reduce the enormous union bound over cuts to a single pair of large vertex sets that sit inside every near-max-cut.
    Direct hypergraph adaptation of the graph notion from DeMarco–Kahn / Hoshen–Samotij–Zhukovskii; existence is proved in §6 rather than postulated.
  • Families Q_1, Q_2, Q_3 of coloured low-/high-degree subgraphs extracted from a near-extremal F-free H no independent evidence
    purpose: Canonical sparse witnesses that force either a larger cut or many edge-disjoint crossing Fanos.
    Proof device internal to §2; existence follows from a probabilistic sparsification (Claim 2.1) and degree case analysis.
  • Sparsified random subfamilies F_q of Fano copies (Lemma 7.3) no independent evidence
    purpose: Restore a usable Δ_p / μ_p ratio when p is large and Q ∈ Q_2 has only moderately small degrees.
    Technical random sparsification; existence is proved by Chernoff + union bound, not assumed.

reviewed 2026-07-31 · how reviews work

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Cite this review

Pith. "Pith review of Random Tur\'an Theorem for the Fano Plane." pith.science (2026). https://pith.science/paper/QSHCHK4Q

@misc{pith2026260728071,
  author       = {Pith},
  title        = {Pith review of: Random Tur\'an Theorem for the Fano Plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QSHCHK4Q}},
  note         = {Machine review of arXiv:2607.28071}
}
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abstract

Let $F$ denote the Fano plane, the $3$-uniform hypergraph with $7$ vertices and $7$ edges. Frankl and F\"uredi, and independently Keevash and Sudakov, proved that the largest $F$-free subhypergraph of $K_n^{(3)}$ is bipartite. In this paper, we determine the sharp threshold for this property in the random setting. We show that for $\hat{p} = \Theta_F \cdot n^{-2/3} \left(\log n\right)^{1/6}$, where $\Theta_F$ is an explicit constant depending on $F$, we have: (i) if $(1+\epsilon) \hat{p} \le p = o(1)$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is bipartite; and (ii) if $\frac{1}{n^2} \ll p \le (1-\epsilon) \hat{p}$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is not bipartite. To the best of our knowledge, this work provides the first sharp threshold result obtained for a Tur\'an-type problem in random hypergraphs.

Figures

Figures reproduced from arXiv: 2607.28071 by Ilay Hoshen.

Figure 1
Figure 1. Figure 1: The 3-uniform hypergraphs F5 and the Fano plane. A hypergraph G is said to be r-partite if its vertices can be partitioned into r sets such that there are no edges fully contained in one of the sets. Further, G is said to be strongly r-partite if its vertices can be partitioned into r sets such that the intersection of every edge with each of the sets is of size at most one. Frankl and F¨uredi [8] establis… view at source ↗
Figure 2
Figure 2. Figure 2: The graph Q[A1] in case Q looks like stars with centres v1, . . . , vk. Also, there are similar stars with the same set of centres but the other vertices are in A2. Proof. First, if ∆1(H[A1]) ⩽ ϵ˜ n 2p log n , then we may take Q = H[A1], which satisfies the first item of the claim. For the remainder of the proof, we assume this is not the case. Let v1, . . . , vk ∈ A1 be the vertices of degree at least 2ηn… view at source ↗
Figure 3
Figure 3. Figure 3: The figure on the left presents a copy of F lying on an edge f ∈ Q with four additional vertices in A2. The blue vertex, together with the two vertices in the first blue set in A2, forms an edge, as does the blue vertex with the two vertices in the second blue set. The same structure applies to the green and red vertices with their corresponding sets of size two in A2 of matching colours. Observe that ther… view at source ↗

discussion (0)

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Reference graph

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This paper was first reviewed by grok-4.5 on July 31, 2026.