{"id":"18c61905-89ac-4d95-b9c2-fe2f162500de","arxiv_id":"2501.19229","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For large n, the unique extremal Δ_r-free r-graph is the balanced complete r-partite r-graph, with a min-degree stability theorem behind it.","lead":"This paper proves a stability version of Mantel's theorem for hypergraphs: any large hypergraph with no triangle substructure and with high minimum degree must be r-partite. This resolves, for large vertex sets, an open question by Mubayi and Pikhurko and identifies the unique densest triangle-free construction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Final-paragraph removal-lemma step counts homomorphic images of Δ_r but cites the subgraph removal lemma; the bridge from Δ_r-free to T_r-free is unproved and load-bearing.","rationale":"The reader's conditional verdict is appropriate, but the load-bearing weakness I find is different from the reader's stated weakest assumption. The reader focused on external dependencies and on the omission when applying Proposition 5.1 to G ∈ H. That application gap is real but appears repairable: if G is G0-colorable for a 2-covered T_r-free G0, then G embeds into the blowup B = G0[V_1,...,V_m] with fibers V_i = φ^{-1}(i), and δ(B) ≥ δ(G); applying Proposition 5.1 to B gives B r-partite and hence G r-partite. My concern is more serious and is located in the same final paragraph: the removal-lemma bridge from Δ_r-free to T_r-free is not justified. The statement 'for every F ∈ T_r there is \\tilde F ∈ Δ_r with a homomorphism \\tilde F → F' points in the wrong direction for the subgraph removal lemma; it explains why a T_r-free graph is Δ_r-free, not why a Δ_r-free graph can be made T_r-free by few edge deletions. Because Theorem 1.2 ultimately derives Δ_r degree stability from T_r edge stability via Theorem 2.3(ii), this step is load-bearing. I do not assert the theorem is false: the needed supersaturation statement may be true, and the author's T_r framework plus the entropy method may be salvageable. But as written, the proof has a genuine gap. The verdict CONDITIONAL remains correct: acceptance should await either a repaired removal argument or an explicit citation of a homomorphism removal lemma whose hypotheses are verified.","tokens_in":15346,"tokens_out":28859,"duration_ms":271248,"concrete_test":"Re-derive the final-paragraph application explicitly. Fix the finite family T_r^{≤2r} = {F ∈ T_r : |V(F)| ≤ 2r} and state which removal lemma applies. If it is applied to \\tilde F ∈ Δ_r, note that Δ_r-freeness gives zero \\tilde F-copies, so the lemma removes no edges and cannot yield T_r-freeness. If it is applied to F ∈ T_r, prove the required hypothesis that every Δ_r-free H has o(n^{|V(F)|}) copies of each such F. For a concrete check, take r = 3 and F = {abc, abd, acd}; attempt to construct a Δ_3-free 3-graph with |H| ≥ (1/9 - o(1)) n^3 and ω(n^3) copies of F. If such a construction exists, the claimed o(n^3)-edge removal fails. If it does not exist, the missing supersaturation proof must be inserted before the step is valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.2 (Section 5, final paragraph), the paper claims that because every F ∈ T_r admits a homomorphism from some \\tilde F ∈ Δ_r, a standard application of the Hypergraph Removal Lemma shows that every Δ_r-free r-graph can be made T_r-free by deleting o(n^r) edges. This does not follow from the cited removal lemma. Hypergraph removal lemmas control subgraph copies of a fixed family; they do not control non-injective homomorphisms. A Δ_r-free H contains zero subgraph copies of \\tilde F, but it may contain many copies of a homomorphic image F ∈ T_r on fewer than |V(\\tilde F)| vertices. For r = 3, F = {abc, abd, acd} is such an image: it has no T_{3,1} subgraph, so a single copy does not violate Δ_3-freeness, yet it is in T_3. If H contains many overlapping copies of such an F, deleting all of them may require Ω(n^r) edge deletions, not o(n^r). No argument is supplied bounding the number of copies of F in a Δ_r-free H, nor is a homomorphism removal lemma stated with verified hypotheses. This step is essential: it converts Δ_r edge-stability into T_r edge-stability and thereby feeds Theorem 2.3(ii) to obtain the degree stability stated in Theorem 1.2. Without it, the central theorem is not established by the written proof.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an Andrásfai–Erdős–Sós-type stability theorem for triangle-free r-uniform hypergraphs: for each r ≥ 2, every sufficiently large Δ_r-free r-graph on n vertices with minimum degree at least n^{r-1}/r^{r-1} − ε n^{r-1} is r-partite, provided ε and n are chosen appropriately. It follows that for large n the balanced complete r-partite r-graph T_r(n) is the unique extremal Δ_r-free construction. This is presented as a stronger form of Mubayi and Pikhurko's Problem 20 on weakly triangle-free r-graphs. The proof combines the entropy–Lagrangian framework of Chao–Yu with the stability framework of Liu–Mubayi–Reiher and Hou–Liu–Zhao, together with a new vertex-extendability argument for T_{r,1} and a symmetrized stability proposition.","tokens_in":15617,"tokens_out":51217,"duration_ms":488563,"significance":"If the proof is completed, the result is significant: it gives the first degree-stability statement for the family Δ_r and settles the extremal uniqueness question for weakly triangle-free r-graphs for large n, in a stronger form than originally asked. The paper is clearly organized, and the entropy computations in Section 3 are carefully executed; Proposition 3.1, in particular, is a clean and nontrivial structural statement. The paper is honest about its reliance on external results, especially Chao–Yu's Lagrangian theorem for T_r-free hypergraphs and the stability framework of [LMR23, HLZ24]. However, two load-bearing steps in the proof of Theorem 1.2 are not justified as written.","major_comments":[{"comment":"The claim that a standard application of the Hypergraph Removal Lemma turns Δ_r-freeness into T_r-freeness after deleting o(n^r) edges is not justified. The usual removal lemma controls subgraph copies of a fixed family, while Δ_r-freeness only forbids subgraphs isomorphic to members of Δ_r, not homomorphisms from those members. A copy of F ∈ T_r in a Δ_r-free graph need not contain any member of Δ_r; for example, for r = 3 the 3-graph F = {abc, abd, acd} belongs to T_3 but has no Δ_3 subgraph, and a star centered at one vertex is Δ_3-free while containing many such F-copies. The text does not supply the required bound on the number of copies of each F in a Δ_r-free graph, nor does it state and verify a homomorphism removal lemma. This step is load-bearing because it converts Δ_r edge-stability into T_r edge-stability before Theorem 2.3(ii) is applied.","section":"Section 5, final paragraph"},{"comment":"Proposition 5.1 is applied to G = H − v* solely because G belongs to the family H. But H is defined as the family of r-graphs that admit a homomorphism to some 2-covered T_r-free r-graph; this does not imply that G is symmetrized, i.e., a full blowup of a 2-covered graph. Proposition 5.1 has symmetrized as an explicit hypothesis. Without an additional argument (for example, a symmetrization procedure preserving the high minimum degree and T_r-freeness), the conclusion that G is r-partite does not follow. This gap affects the proof that T_r is vertex-extendable with respect to H, which is needed for the invocation of Theorem 2.3(i).","section":"Section 5, proof of Theorem 1.2"}],"minor_comments":[{"comment":"There are a few harmless typographical slips, such as 'max' used instead of a set in the definition of S_{n−1} in Section 2, and 'Proposition 1.2' instead of 'Theorem 1.2' at the end of the proof in Section 5.","section":"Throughout"},{"comment":"The fact that a 2-covered T_r-free r-graph is a (v(H), r, r−1)-system is used repeatedly; it may be worth adding a one-sentence proof or a precise reference, since it is a key structural input.","section":"Section 2, Fact 2.2(ii)"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the overall strategy is attractive, but the two gaps identified in the major comments are both in the proof of the central theorem. I believe they are repairable, but the written proof is not yet complete. The paper also relies heavily on two framework papers co-authored by the author; this is not a problem per se, but the editor may want the revised version to state clearly which ingredients are imported and which are new."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper claims an Andrásfai–Erdős–Sós-type stability theorem for hypergraph triangles: for large n, every Δ_r-free r-graph with minimum degree close to n^{r-1}/r^{r-1} is r-partite, and T_r(n) is the unique extremal construction. This is a significant result if it holds; it resolves the large-n case of Mubayi–Pikhurko's Problem 20 and extends prior work for r=2,3 to all r. The entropy method from Chao–Yu is combined with the LMR/HLZ stability framework, and the Lagrangian uniqueness proposition (Prop 3.1) is a solid piece of work on its own.\n\nThe proof, however, is not fully written. Two gaps stand out. First, in the vertex-extendability argument, Proposition 5.1 is applied to G = H - v* merely because G belongs to the hereditary family H. Membership in H gives a homomorphism from G to some 2-covered T_r-free graph; it does not make G symmetrized, which is what Proposition 5.1 requires. This is a missing step, not a contradiction, and it might be repairable.\n\nSecond, the final paragraph claims that every Δ_r-free r-graph can be made T_r-free by deleting o(n^r) edges, via a standard application of the Hypergraph Removal Lemma. That lemma controls subgraph copies; it does not control the homomorphic images that define T_r. Indeed, for r=3, the blowup of the 4-vertex 3-graph {abc, abd, acd} is Δ_3-free but requires Θ(n^3) edge deletions to destroy all copies of that T_3 member. So the statement is false as written. The argument needs a version for near-extremal Δ_r-free graphs, and none is given. This step is load-bearing: it is what converts T_r edge-stability into Δ_r degree-stability.\n\nThe rest of the paper—the entropy/Lagrangian calculations, the compactness argument in Prop 5.1, the auxiliary STS application—reads as careful and technically sound. The citation to [CY24] for the Lagrangian equality is appropriate, despite the heavy self-citation of the stability framework.\n\nWho is this for? Anyone working in hypergraph Turán problems. The theorem is likely true and important, but the current version is not the one to rely on. I'd send it to a strong referee, but with a clear request to check the two gaps above; they are not fatal to the approach, but they are not cosmetic either.\n\nRecommended action: engage, but ask for a revised proof of the removal step.","headline":"A plausible and important stability theorem for hypergraph triangles, but the proof as written has two gaps—one in the vertex-extendability step and one in the homomorphic-deletion step—that need repair before the result is established.","tokens_in":16198,"tokens_out":17037,"would_cite":true,"duration_ms":146395,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for each r ≥ 2, every sufficiently large triangle-free r-graph with minimum degree within o(n^{r-1}) of the Turán threshold is r-partite, making the balanced complete r-partite hypergraph the unique extremal…","keywords":["hypergraph Turán problem","triangle-free hypergraphs","stability theorem","Lagrangian","entropy","degree stability","extremal hypergraphs","Andrásfai–Erdős–Sós theorem"],"falsifier":"Exhibit, for some r ≥ 4, a Δ_r-free r-graph on arbitrarily large n with minimum degree at least $n^{{r-1}}$/$r^{{r-1}}$ − o($n^{{r-1}}$) that is not r-partite; this would directly contradict Theorem 1.2. A cheaper check is to compute the Lagrangian of a candidate 2-covered T_r-free hypergraph and see whether it exceeds 1/r^r, which would falsify Proposition 3.1.","tokens_in":15116,"feed_emoji":"🔺","tokens_out":6483,"duration_ms":57097,"temperature":0.7,"pith_summary":"The paper proves an Andrásfai–Erdős–Sós-type stability theorem for triangle-free r-uniform hypergraphs: for every r ≥ 2, any sufficiently large hypergraph that avoids the family Δ_r and has minimum degree close to (1/r)^{r-1} $n^{{r-1}}$ must be r-partite. The direct consequence is that, for large n, the balanced complete r-partite r-graph T_r(n) is the unique extremal construction among Δ_r-free hypergraphs on n vertices. This answers an open problem about weakly triangle-free r-graphs in a stronger form than was asked. The proof gets its key input from an entropic reformulation of the hypergraph Lagrangian and from a general stability framework that upgrades edge stability to degree stability.","feed_headline":"Large triangle-free hypergraphs are forced r-partite","feed_subtitle":"Balanced complete r-partite hypergraph is the unique extremal construction for large n.","key_machinery":"The load-bearing object is the hypergraph Lagrangian, the maximum of the polynomial P_H(x) = Σ_{e∈H} ∏_{i∈e} x_i over the simplex, together with its entropic reformulation: the entropy density equals r! λ(H). Proposition 3.1 shows that for a 2-covered triangle-free r-graph the only optimal weight vector is uniform on a single edge, (1/r,...,1/r,0,...,0). This uniqueness is combined with vertex-extendability of the triangle configuration with respect to r-partite hypergraphs and a stability framework that derives degree stability from edge stability plus vertex-extendability.","core_discovery":"The central claim is Theorem 1.2: there exist ε(r) > 0 and N0(r) such that every Δ_r-free r-graph H on n ≥ N0(r) vertices with δ(H) ≥ $n^{{r-1}}$/$r^{{r-1}}$ − ε $n^{{r-1}}$ is r-partite. For large n this makes T_r(n) the unique extremal Δ_r-free construction, and the same argument yields the exact bound |H| ≤ n^r/r^r, with equality exactly for T_r(n) when r divides n. The theorem is stronger than the weakly triangle-free problem it answers, because Δ_r-freeness is a weaker hypothesis than weakly triangle-free.","pith_inferences":["Editorial inference: the Lagrangian-uniqueness proposition likely holds for other 'sharp' hypergraph families, so the same proof scheme should yield degree-stability theorems for families whose Lagrangian is known to be 1/r^r and uniquely attained on one edge.","Editorial inference: the optimal stability window ε(r) is left open; a natural conjecture is that for r=3 the largest ε matches the exact threshold from the recently solved case, so the stability regime may be as wide as possible.","Editorial inference: since the proof only needs the Lagrangian value and uniqueness, the method may transfer to L-intersecting families with L = [i] and i < ⌈r/2⌉, where the same Lagrangian bound holds, potentially answering part of Problem 6.2."],"forward_implications":["For large n, T_r(n) is the unique extremal Δ_r-free construction, settling the large-n case of Mubayi–Pikhurko's Problem 20.","Combined with a standard blow-up argument, it gives the exact Turán number ex(n, Δ_r) = n^r/r^r for large n, with equality only for T_r(n).","Together with known results on spectral Turán problems, it solves the α-spectral Turán problem for Δ_r for all α ≥ 1 and large n.","It yields a generalized Turán theorem for copies of Steiner triple systems in C3-free 3-graphs: the extremal number is |T_k(n)|.","The same stability route applies to the single hypergraph described in [CY24, Theorem 1.5] with only minor modifications."],"supporting_citations":[{"why":"Supplies the entropic technique and the key theorem that every T_r-free r-graph has Lagrangian 1/r^r, which fixes the value β in Proposition 3.1 and bounds links.","marker":"[CY24]"},{"why":"Provides the vertex-extendability framework used to upgrade symmetrized stability to degree stability.","marker":"[LMR23]"},{"why":"Refines the stability framework and provides the compactness argument used to pass from Lagrangian uniqueness to symmetrized stability.","marker":"[HLZ24]"},{"why":"States Problem 20 on weakly triangle-free r-graphs, the question Theorem 1.2 answers for large n.","marker":"[MPS11]"},{"why":"Bridges the stability result to the α-spectral Turán problem for Δ_r.","marker":"[KLM14]"},{"why":"Supplies the spectral-radius result used with Theorem 1.2 to settle the α-spectral Turán problem.","marker":"[KNY15]"}],"fun_headline_variants":["Triangle-free r-graphs are r-partite for large n","Stability theorem yields unique extremal hypergraph","Hypergraph Mantel: stability forces r-partite","Unique extremal triangle-free hypergraph is r-partite","Stronger form of weak triangle-free problem solved"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument leans on an imported result that has not been re-proved here: every triangle-free hypergraph has a certain maximum-polynomial value equal to 1/r^r; if that result were wrong, the degree-stability conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Triangle-free r-graphs are r-partite for large n","Stability theorem yields unique extremal hypergraph","Hypergraph Mantel: stability forces r-partite","Unique extremal triangle-free hypergraph is r-partite","Stronger form of weak triangle-free problem solved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000523,"raw_usage":{"total_tokens":2515,"prompt_tokens":921,"completion_tokens":1594,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":1516}},"tokens_in":537,"tokens_out":1594,"duration_ms":12559,"temperature":1.0,"reasoning_tokens":1516,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T20:52:59.063852+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit, for some r ≥ 4, a Δ_r-free r-graph on arbitrarily large n with minimum degree at least $n^{{r-1}}$/$r^{{r-1}}$ − o($n^{{r-1}}$) that is not r-partite; this would directly contradict Theorem 1.2. A cheaper check is to compute the Lagrangian of a candidate 2-covered T_r-free hypergraph and see whether it exceeds 1/r^r, which would falsify Proposition 3.1.","supporting_citations":[],"review_version":1}