{"id":"3ab33dad-7a5e-4e16-81ce-1097c407cd69","arxiv_id":"2602.14008","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Oriented graphs that exceed the oriented Turán density contain a positive fraction of the possible copies of the forbidden oriented subgraph, with explicit bounds for transitive tournaments and antidirected complete bipartite graphs.","lead":"Extremal graph theory asks how many edges force a pattern; this paper answers the oriented version: once an oriented graph has more arcs than the extremal threshold, it must contain many copies of the forbidden oriented pattern. It proves an oriented Erdős–Simonovits supersaturation theorem, a Moon–Moser type inequality for transitive tournaments, and explicit copy-count bounds for antidirected complete bipartite graphs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.9 double-counting lower bound rests on a false claim: a non-transitive (r+1)-tournament may lack a vertex y such that every r-subset containing u' and y is non-transitive.","rationale":"The reader's conditional verdict is based on precisely this flaw in the proof of Theorem 1.9, and my analysis confirms it. The counterexample is simple and destroys the 'some y' assertion as written. The gap is load-bearing because the lower bound on P drives the Moon–Moser inequality and the supersaturation bound for transitive tournaments. The other central results (Theorem 1.7, Theorem 1.8, Theorem 1.11) do not depend on this step. No new concern beyond the reader's is identified, and no reason to change the verdict appears: the paper should remain conditional pending a repair or replacement of the disputed double-counting argument.","tokens_in":14082,"tokens_out":4943,"duration_ms":41280,"concrete_test":"Directly test the disputed assertion on the r=3 counterexample: let G be the 4-vertex tournament with cyclic triangle 1→2→3→1 and dominating vertex 4 (4→1,4→2,4→3). Take Q_j={1,2,4} and u'=3, the unique non-extension. For each y∈{1,2,4}, enumerate the r−1=2 triples containing u', y, and one other vertex of Q_j\\{y}; verify that in every case at least one triple is transitive, so no y works. Then recompute the lower bound P ≥ (r−1)(N_3(4−3)−4N_4) to see it does not follow from the asserted intermediate claim. If the authors replace the claim with a valid argument, re-check Theorem 1.9; otherwise the Moon–Moser inequality remains unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4, proof of Theorem 1.9: the lower bound P ≥ (r-1) Σ_j (n-r-b_j) is derived from the assertion that for every non-extension u' of a transitive Q_j there is y∈V(Q_j) such that u', y together with any r−2 other vertices of Q_j\\{y} induce a non-transitive r-set. This assertion is false. For r=3, take G on {1,2,3,4} with arcs 1→2, 2→3, 3→1 and 4→1,4→2,4→3. The only non-transitive triple is {1,2,3}; the triples {1,2,4}, {1,3,4}, {2,3,4} are transitive. Let Q_j={1,2,4} and u'=3. For y=1, {1,3,4} is transitive; for y=2, {2,3,4} is transitive; for y=4, both triples containing 3 and 4 are transitive. Thus no y satisfies the claimed property. Since Theorem 1.9 and its corollary Theorem 1.10 rely on this count, the proof has a genuine gap. Theorems 1.7, 1.8, and 1.11 are unaffected. This is the same load-bearing weakness identified by the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops supersaturation results for oriented Turán problems. Theorem 1.7 gives a general Erdős–Simonovits-type statement: any oriented graph with edge density above the oriented Turán density of a fixed oriented graph F contains a positive-density number of copies of F. Theorem 1.8 establishes a Rademacher-type bound for transitive triangles: an n-vertex oriented graph with one more arc than the balanced complete 3-partite oriented extremal graph contains about 2n/3 copies of TT3. Theorem 1.9 claims a Moon–Moser-type inequality for transitive tournaments TTr, and Theorem 1.10 derives a supersaturation bound for general r from it. Theorem 1.11 gives a supersaturation result for antidirected complete bipartite graphs K_{s,t}. The proofs of Theorems 1.7, 1.8, and 1.11 are self-contained averaging/induction arguments; the proof of Theorem 1.9, however, contains a false intermediate claim that is load-bearing for both Theorems 1.9 and 1.10.","tokens_in":14434,"tokens_out":2717,"duration_ms":23568,"significance":"If all the results were correct, this would be a meaningful step for oriented Turán supersaturation, especially the first oriented analogue of the Moon–Moser inequality and the explicit supersaturation bound for antidirected complete bipartite graphs. Theorem 1.7 is a clean and correct convergence/density argument, and Theorem 1.8 is a carefully detailed induction with only local technical points. Theorem 1.11 gives explicit constants and is also essentially correct. The main defect is the proof of Theorem 1.9: the double-counting lower bound relies on a false structural assertion, and because Theorem 1.10 is derived from Theorem 1.9, the gap propagates. The paper’s central claims are therefore currently unproven in their full stated generality, though the affected theorem may be repairable. The manuscript is self-contained and does not appear to depend improperly on the authors’ prior work.","major_comments":[{"comment":"The lower bound P ≥ (r−1) Σ_j (n−r−b_j) is derived from the assertion: for every non-extension u' of a transitive r-tournament Q_j, there is a vertex y∈V(Q_j) such that u', y together with any r−2 other vertices of Q_j\\{y} induce a non-transitive r-set. This assertion is false. For r=3, take the oriented graph G on {1,2,3,4} with arcs 1→2, 2→3, 3→1, and 4→1, 4→2, 4→3. The only non-transitive triple is {1,2,3}. Let Q_j={1,2,4} and u'=3. For y=1, the triple {1,3,4} is transitive; for y=2, {2,3,4} is transitive; for y=4, both {3,4,1} and {3,4,2} are transitive. Thus no y has the claimed property. Consequently, the double-counting inequality P ≥ (r−1)Σ_j(n−r−b_j) is not justified. Since this inequality is used to derive the Moon–Moser inequality in Theorem 1.9 and then Theorem 1.10, those two theorems are not proven by this manuscript. The remaining results (Theorems 1.7, 1.8, 1.11) are unaf","section":"Section 4, proof of Theorem 1.9"},{"comment":"Theorem 1.10 is derived from Theorem 1.9 via Claim 4.1. Because the proof of Theorem 1.9 has the gap described above, the proof of Theorem 1.10 inherits that gap. The statement of Theorem 1.10 may still be true, but a different argument or a corrected double-counting inequality is needed to establish it.","section":"Section 4, Theorem 1.10"}],"minor_comments":[{"comment":"In the double-counting display, the subscript 's−1' appears in 'Ns−1' where it should be 'r−1'. This is a typo that should be corrected.","section":"Section 4, proof of Theorem 1.9"},{"comment":"The notation π_o(F) is defined as a limit of ex_o(n,F)/binom(n,2). The limit is well-defined by Proposition 2.1, but it would be clearer to write the limit explicitly as n→∞ in the definition.","section":"Section 1, Definition 1.2"},{"comment":"Reference [17] (Taylor) appears in the bibliography but is not cited in the body of the paper. Either cite it where relevant (e.g., in the discussion of regular methods for digraphs) or remove it.","section":"References"},{"comment":"There is a small punctuation typo: 'by Claim 3.2,.' should read 'by Claim 3.2.'.","section":"Section 3, Case 3.2.1"}],"recommendation":"major_revision","confidential_remarks":"The false claim in the proof of Theorem 1.9 is a genuine load-bearing gap, but I do not think this warrants rejection outright. The theorem may be repairable, and the unaffected results (Theorems 1.7, 1.8, 1.11) are solid and interesting. However, the authors must either fix the double-counting lower bound or clearly isolate which results can be salvaged. I would also ask them to check whether the Moon–Moser inequality itself, without this proof, is already known or can be proven by another method."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is not a package deal. The oriented Erdős–Simonovits theorem (1.7), the Rademacher-type result for transitive triangles (1.8), and the K_{s,t} bound (1.11) are all new and, as far as I can tell, correctly proved. The induction in 1.8 is messy but the cases check out. The averaging in 1.7 is clean. The algebra in 1.11 is fine. If the paper only contained these, I would send it out with confidence.\n\nThe problem is Theorem 1.9, the Moon–Moser type inequality, and consequently Theorem 1.10. The double-counting lower bound on P is not justified. The proof asserts that for every non-extension u' of a transitive r-tournament Q_j, there is a vertex y in Q_j such that u', y together with any r−2 other vertices of Q_j\\{y} form a non-transitive r-set. That assertion is false. For r=3, take the tournament on {1,2,3,4} with arcs 1→2, 2→3, 3→1, and 4→1,4→2,4→3. The only non-transitive triple is {1,2,3}. Let Q_j={1,2,4} (transitive) and u'=3. For y=1,2,4, every triple containing 3 and y is transitive. So no such y exists. The lower bound P ≥ (r−1)Σ(n−r−b_j) rests on this claim, and the Moon–Moser inequality and Theorem 1.10 fall with it. This is not a minor typo; the induction in Theorem 1.10 depends on it.\n\nThe counterexample also shows the issue is not just an awkward ordering of cases. It is a structural obstacle: a non-transitive (r+1)-tournament can have exactly one non-transitive r-subset. I don't see an easy repair in the current proof. Maybe a broader family of r-subsets, or a different count, could salvage the result, but as written the general supersaturation for transitive tournaments is unproven.\n\nThe reader's report and the stress-test note are on target. I would not desk-reject: the paper is worth a serious referee, and the unaffected results deserve to be published. But the acceptance should be conditional on fixing 1.9 or dropping it.","headline":"Solid partial paper: three theorems are good, but the Moon–Moser inequality for transitive tournaments has a false double-counting claim and should not be accepted as is.","tokens_in":14883,"tokens_out":3798,"would_cite":false,"duration_ms":32714,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C20","05C35","05C42"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes oriented supersaturation: arc density above the oriented Turán threshold forces a positive-density supply of every forbidden oriented subgraph, including an exact one-extra-arc bound for transitive triangles and a rati","keywords":["oriented Turán problem","supersaturation","transitive tournament","oriented Turán density","antidirected complete bipartite graph","transitive tournament counting","extremal oriented graphs","Moon-Moser inequality"],"falsifier":"The lower bound in the double count assumes that for each transitive r-set Q and each vertex u' whose addition does not make a transitive (r+1)-set, some vertex y in Q makes u', y together with any other r−2 vertices of Q non-transitive. This is false: in the four-vertex tournament with directed triangle a→b→c→a and d dominating a,b,c, take the transitive triple Q={a,b,d} and u'=c; no choice of y gives the claimed property for all remaining vertices. That failure means Theorem 1.9's ratio inequality needs a repaired argument.","tokens_in":13977,"feed_emoji":"📈","tokens_out":8706,"duration_ms":66284,"temperature":0.7,"pith_summary":"Turán-type extremal problems ask how many arcs an oriented graph can have while avoiding a fixed oriented pattern. This paper tries to establish the supersaturation side of that question: once the arc density passes the extremal threshold, the forbidden pattern should appear not just once but in positive proportion. The central claims are a general positive-density statement for every fixed oriented graph, an exact one-extra-arc count for the transitive triangle, a ratio inequality linking counts of transitive tournaments of consecutive orders, and a supersaturation bound for antidirected complete bipartite graphs. One step in the transitive-tournament double count uses an assumption about non-extensions that is not true in general, so the ratio inequality and its corollaries currently depend on an unproved strengthening.","feed_headline":"One extra arc forces about 2n/3 transitive triangles","feed_subtitle":"Cross the oriented extremal threshold and the forbidden pattern appears in positive density—not just once.","key_machinery":"The transitive tournament is the workhorse: the acyclic tournament whose vertices can be ordered so every arc points forward. The general supersaturation proof averages over m-vertex induced subgraphs: if too few m-sets were above the extremal density, the total arc count would fall short, while each above-threshold m-set must contain a copy of F; double-counting m-sets against copies of F yields δ. For transitive tournaments, the argument is a pair count of (T,R) where T is a transitive r-tournament and R is a non-transitive r-set sharing r−1 vertices with T. An upper bound on these pairs via Jensen's inequality and a lower bound obtained by counting non-extensions are combined to produce t","core_discovery":"For a fixed oriented graph F, let π_o(F) be the limit of ex_o(n,F)/(n choose 2). The paper's main theorem says that for every ε>0 there is δ>0 such that every sufficiently large oriented graph with at least (π_o(F)+ε)(n choose 2) arcs contains at least δ(n choose h) copies of F. For the transitive triangle, one arc above the balanced complete 3-partite extremal graph forces roughly 2n/3 copies. With N_r the number of transitive r-tournaments, it derives the ratio inequality N_{r+1}/N_r ≥ (r^2 N_r/N_{r-1} − n)/(r^2−1), and from it a supersaturation bound whenever |E(G)| ≥ (1−1/t)n^2/2. It also shows that |E(G)| ≥ e s^{1/t} n^{2−1/t} forces at least (e/t)^t n^t copies of the antidirected compl","pith_inferences":["The faulty lower-bound step suggests Theorem 1.9 may still be salvageable by counting only the non-transitive r-sets that do satisfy the required property, possibly with a weaker constant; a natural test is to compute the inequality on the four-vertex counterexample and see which term breaks.","If a corrected ratio inequality holds, it would imply a local counting stability for transitive tournaments: tournaments with many transitive r-sets must contain many transitive (r+1)-sets, a directed analogue of clique supersaturation that could feed into extremal results for acyclic oriented graphs.","The exact one-extra-arc result for the transitive triangle invites a full stability-supersaturation classification for all tournaments of order three, including the cyclic triangle, whose extremal behaviour differs because oriented graphs containing a directed cycle can hide copies differently."],"forward_implications":["If the main theorem holds, an n-vertex oriented graph with arc density π_o(F)+ε contains at least a fixed positive fraction δ of all possible h-vertex sets as copies of F; the threshold is truly a phase transition.","For transitive triangles, one arc beyond the balanced complete 3-partite extremal graph forces at least about 2n/3 copies—an oriented analogue of the sharp triangle-counting result for graphs.","The ratio inequality N_{r+1}/N_r ≥ (r² N_r/N_{r-1} − n)/(r²−1) links consecutive transitive-tournament counts; if valid, it yields the supersaturation bound N_r ≥ (t choose r)(n/t)^r for oriented graphs with arc count at least (1−1/t)n²/2.","For antidirected complete bipartite graphs, arc count ≥ e s^{1/t} n^{2−1/t} forces polynomial many copies, extending bipartite supersaturation to oriented settings."],"fun_headline_variants":["One extra arc forces ~2n/3 transitive triangles","Supersaturation for oriented Turán: cross threshold, copies appear","Oriented Turán supersaturation: a single arc triggers many copies","Transitive triangles multiply: threshold +1 arc = ~2n/3 copies","New proof: Turán threshold crossing yields positive pattern density"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The transitive-tournament ratio rests on the assumption that whenever a vertex u' fails to extend a transitive r-tournament Q, some vertex y in Q has the property that u', y together with any r−2 other vertices of Q is non-transitive; that 'any' fails even for r=3, so the inequality as proved needs a repaired count.","fun_headline_variants_meta":{"raw":{"variants":["One extra arc forces ~2n/3 transitive triangles","Supersaturation for oriented Turán: cross threshold, copies appear","Oriented Turán supersaturation: a single arc triggers many copies","Transitive triangles multiply: threshold +1 arc = ~2n/3 copies","New proof: Turán threshold crossing yields positive pattern density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1315,"prompt_tokens":694,"completion_tokens":621,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":531}},"tokens_in":438,"tokens_out":621,"duration_ms":6328,"temperature":1.0,"reasoning_tokens":531,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T23:30:51.463210+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The lower bound in the double count assumes that for each transitive r-set Q and each vertex u' whose addition does not make a transitive (r+1)-set, some vertex y in Q makes u', y together with any other r−2 vertices of Q non-transitive. This is false: in the four-vertex tournament with directed triangle a→b→c→a and d dominating a,b,c, take the transitive triple Q={a,b,d} and u'=c; no choice of y gives the claimed property for all remaining vertices. That failure means Theorem 1.9's ratio inequality needs a repaired argument.","supporting_citations":[],"review_version":1}