{"id":"93e26672-acb8-4fd5-a6d5-bb731379b28a","arxiv_id":"2607.28071","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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}.","lead":"This paper finds the exact probability threshold where the largest Fano-plane-free piece of a random 3-uniform hypergraph switches from non-bipartite to bipartite. It is the first sharp threshold of this Turán type proved for random hypergraphs.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the stability black box as the main external dependency and correctly judges that it sits inside its published range once p is above the sharp threshold. No tighter load-bearing failure appears on a second pass: the Fano-specific counting (linear, strictly 3-balanced, every pair in one edge) is used consistently in Lemmas 5.1–5.2 and 9.1–9.2; the hypergraph rigidity adaptation tracks the graph proofs with adjusted exponents that close; and the 0-statement second-moment comparison is calibrated exactly to the definition (2) of Θ_F. Ordinary verification risk remains, but nothing that would move the verdict from ACCEPT. Concrete test above is a low-cost sanity check on the only slightly loose point (fixed vs vanishing eta), not a threat to the claim.","tokens_in":53372,"tokens_out":719,"duration_ms":55080,"concrete_test":"Fix the constant hierarchy of §2.1 with a concrete small eta (e.g. eta=10^{-3}) and the explicit Θ_F from (1)–(2); verify that for this fixed eta the C(eta) of Conlon–Gowers is absorbed by the (log n)^{1/6} factor for all n large enough that (1+ε)Θ_F (log n)^{1/6} > C(eta), and that Claim 2.4’s lower bound d_Q>2e(H[A_1]) continues to hold under the resulting numerical values of η,ε̃. If it does, the stability invocation is secure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.3) rests on a complete, self-contained adaptation of the graph-case sharp-threshold architecture (stability \to Q-extraction \to fixed-cut Janson \to rigidity switching \to second-moment 0-statement) to the Fano plane. The external input flagged by the reader—Conlon–Gowers stability (Theorem 1.4)—applies directly: for any fixed eta>0 small enough for the constant hierarchy of §2.1, there is C(eta) such that p⩾(1+ε)Θ_F n^{-2/3}(log n)^{1/6} eventually exceeds C(eta)n^{-2/3}, and the strict inequality e(H∩int)<eta n^{3}p supplies the needed room in Claim 2.4. The paper’s writing of eta=eta(n)≪1 is slightly loose (a fixed small eta suffices), but does not create a gap. Rigidity (Thm 6.4/Cor 6.5), the low/high-degree Janson estimates (Lemmas 5.1–5.2), the switching of Lemma 7.2, and the 0-statement second-moment argument are written out in full with hypergraph-adjusted error terms. Residual risk is ordinary long-proof arithmetic/intersection-case risk, not a structural soft spot in the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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.","tokens_in":53715,"tokens_out":1312,"duration_ms":35279,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§2.1 and after (6)"},{"comment":"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.","section":"throughout"},{"comment":"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).","section":"§1, (1)"},{"comment":"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.","section":"§2, Claim 2.2"},{"comment":"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.","section":"§7.2–7.3"},{"comment":"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.","section":"§1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is long and technical but the architecture is sound and the result is genuinely the first of its kind in random hypergraphs. I see no load-bearing gap; the Conlon–Gowers input applies cleanly at the threshold scale. Minor revision for copy-editing and a few clarity notes is appropriate; I would not ask for a major restructuring. Fit for a top combinatorics journal is good."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the first sharp two-sided threshold for a Turán-type property in random hypergraphs. Hoshen pins the Fano plane in G_{n,p}^{(3)} at the explicit constant Θ_F n^{-2/3}(log n)^{1/6}: above it every largest F-free subhypergraph is bipartite whp, below it none is. That is the natural random analogue of Frankl–Füredi / Keevash–Sudakov, and prior hypergraph work only had coarse windows for F_5.\n\nWhat is actually new is the full transfer of the graph sharp-threshold programme (stability → Q-extraction → fixed-cut Janson → rigidity/core → switching that kills the cut union bound → second-moment 0-statement) to 3-graphs. The low-degree and high-degree Janson estimates (Lemmas 5.1–5.2), the rigidity theorem for hypergraphs (Thm 6.4), the switching of Lemma 7.2, and the resampling argument for the 0-statement are written out with the necessary hypergraph error terms. Θ_F is defined combinatorially from π_F, not fitted. Self-citations supply technique, not the threshold value.\n\nThe load-bearing external input is Conlon–Gowers stability. It applies in the range needed: once p is a constant factor above n^{-2/3}, you get the β-room for the deficit-versus-matching argument. The paper’s wording “β(n)≪1” is slightly loose (a fixed small η works), but that is cosmetic. Residual risk is ordinary long-proof arithmetic and intersection-case bookkeeping, not a structural hole.\n\nThis is for people who already care about random extremal hypergraph theory or who want a template for other hypergraphs with known extremal structure. It deserves a serious referee. I would bring it to reading group, cite it if I work near this area, and send it out for review.","headline":"First genuine sharp Turán threshold in random hypergraphs; the Fano case is fully worked and the architecture looks transferable.","tokens_in":54423,"tokens_out":505,"would_cite":true,"duration_ms":18934,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C80","05D05"],"pacs":[],"model":"grok-4.5","headline":"The largest Fano-plane-free piece of a random 3-uniform hypergraph is bipartite exactly above an explicit density threshold.","keywords":["Fano plane","random hypergraphs","Turán theorem","sharp threshold","stability","cores","3-uniform hypergraphs"],"falsifier":"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.","tokens_in":54195,"feed_emoji":"📐","tokens_out":1061,"duration_ms":21821,"temperature":0.7,"pith_summary":"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.","feed_headline":"Fano-free random hypergraphs turn bipartite at explicit density","feed_subtitle":"First sharp Turán threshold in random 3-graphs sits at Θ n^{-2/3}(log n)^{1/6}","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Sharp threshold forces Fano-free random 3-graphs bipartite","Fano Turán property flips at Θ n^{-2/3}(log n)^{1/6}","Largest Fano-free subhypergraphs bipartite above explicit p","First sharp random hypergraph Turán threshold for Fano plane","Bipartite takeover in Fano-free G_{n,p}^{(3)} at precise density"],"cache_read_input_tokens":49280,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp threshold forces Fano-free random 3-graphs bipartite","Fano Turán property flips at Θ n^{-2/3}(log n)^{1/6}","Largest Fano-free subhypergraphs bipartite above explicit p","First sharp random hypergraph Turán threshold for Fano plane","Bipartite takeover in Fano-free G_{n,p}^{(3)} at precise density"]},"model":"grok-4.5","effort":"low","cost_usd":0.00428,"raw_usage":{"total_tokens":1309,"prompt_tokens":839,"num_sources_used":0,"completion_tokens":108,"cost_in_usd_ticks":42804000,"prompt_tokens_details":{"text_tokens":839,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":362,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":839,"tokens_out":108,"duration_ms":7988,"temperature":1.0,"reasoning_tokens":362,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T18:43:53.118044+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}