{"id":"5bfc4f2f-1cf4-464d-8fdb-d66c94559ec3","arxiv_id":"2507.04579","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For n≥30(s+1), every F5-free 3-graph with matching number at most s has at most s⌊(n−s)^2/4⌋ edges, with equality only for the complete 3-partite hypergraph H3(n,s).","lead":"This paper determines the largest possible number of triples in a 3-uniform hypergraph on n vertices that contains no generalized triangle F5 and has matching number at most s. For n at least 30(s+1) and s at least 3, the answer is s times floor((n-s)^2/4), achieved by a complete 3-partite hypergraph with one part of size s.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Case 2 of Theorem 1.6 contains an invalid algebraic inequality: positive terms of order n are replaced by order-1 terms, so the main upper bound for s≥3 does not follow as written.","rationale":"The reader's weakest_assumption identifies exactly the spot I find load-bearing, and my independent read agrees. The bound in Theorem 1.6 for s≥3 relies on Case 2 for intermediate i, and the displayed inequality there is not arithmetically valid as printed: the previous line has positive terms of order n, while the next line drops the factor n from those terms, making the inequality go in the wrong direction. This is not an aesthetic objection; it is the step that converts the sum of link sizes into a bound below s⌊(n−s)^2/4⌋. The paper has substantial structure that is likely correct: Lemma 3.3, the s≤2 case, and Theorem 2.2 appear coherent, and the final equality case follows if Theorem 1.8's equality statement is supplied. The flaw appears typographical rather than conceptual, since a corrected grouping still yields a negative quadratic dominated by −i(n−s)^2/4 for n≥30s. But the manuscript should not be accepted as is because the central proof contains a false inequality. I also note the equality case of Theorem 1.8 is under-justified; this does not change the main upper bound but is relevant to uniqueness. The appropriate verdict remains conditional pending verification of the algebra.","tokens_in":15233,"tokens_out":15763,"duration_ms":158368,"concrete_test":"Recompute the Case 2 bound symbolically: from (4.1), expand E = (s−i)⌊(n−s)^2/4⌋ + binom(s−i,2)n + 6i^2n + isn, and test whether E ≤ s⌊(n−s)^2/4⌋ for n=30(s+1), i in {s/2, s−2} (with s/2 rounded as in the paper) and s=3,...,100. Then test the corrected identity E = s⌊(n−s)^2/4⌋ + (1/2 s(s−1) + 13/2 i^2 + i/2)n − i⌊(n−s)^2/4⌋. If the printed f(n,s,i) is not an upper bound but the corrected expression is, the flaw is a fixable typo; if the corrected expression also fails, Case 2 is genuinely unproven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.6, Case 2 (s/2 ≤ i ≤ s−2), the displayed chain after applying Theorem 1.8 is invalid. From (4.1) the proof obtains\n\n|H| ≤ s⌊(n−s)^2/4⌋ + 1/2 s(s−1)n + i(13/2 i n + n/2 − ⌊(n−s)^2/4⌋)\n\nand then asserts this is at most\n\ns⌊(n−s)^2/4⌋ + s(s−1) + 13i^2 + i/2 n − i(n−s)^2/4 + i.\n\nThe asserted inequality replaces the positive terms 1/2 s(s−1)n and 13/2 i^2 n by s(s−1) and 13i^2, losing a factor n. For n ≥ 30(s+1), 1/2 s(s−1)n is much larger than s(s−1), so the right-hand side is smaller, not larger, than the preceding expression. The subsequent definition of f(n,s,i) and the estimates f(n,s,s−2)<0 and f(n,s,s/2)<0 rest on this invalid bound. Because this is the only estimate in Case 2 that forces |H| below s⌊(n−s)^2/4⌋, the upper bound in Theorem 1.6 is not established as written. A secondary gap appears in the proof of Theorem 1.8's equality statement: the claim that a triangle in G_i contradicts (2.1) is asserted without the needed derivation; this affects the uniqueness part of Theorem 1.6.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the hypergraph Turán problem for the generalized triangle F5 (a 3-graph on five vertices with edges abc, abd, cde) under an upper bound on the matching number. The main result (Theorem 1.6) states that for n ≥ 30(s+1), an F5-free 3-graph on n vertices with matching number at most s has at most s⌊(n−s)^2/4⌋ edges for s ≥ 3, and at most binom(n−1,2) edges for s = 1,2, with the full star (s=1,2) and the complete 3-partite graph H3(n,s) (s≥3) as the unique extremal hypergraphs. The proof introduces a 2-colored version of Mantel's theorem (Theorem 1.8) as a key tool, applies a structural lemma (Lemma 3.3) that bounds the hypergraph in terms of link degrees, and then proceeds by case analysis on a parameter i. The paper also proposes two conjectures in the concluding remarks.","tokens_in":15609,"tokens_out":8231,"duration_ms":76446,"significance":"If correct, the result is a meaningful addition to the growing literature on hypergraph Turán problems with bounded matching number, extending the graph-theoretic Alon–Frankl theorem and the hypergraph results of Gerbner, Tompkins, and Zhou. The 2-colored Mantel theorem (Theorem 1.8) is a natural and potentially reusable extremal statement about edge-colored graphs, and its proof is attempted from first principles via induction. The paper is self-contained and the high-level proof strategy is coherent, but as discussed below a load-bearing algebraic step in the case analysis of Theorem 1.6 is invalid, and the equality case of Theorem 1.8 is not fully justified. These issues need to be addressed before the main theorem can be considered proven.","major_comments":[{"comment":"The displayed chain of inequalities after applying Theorem 1.8 claims that |H| ≤ s⌊(n−s)^2/4⌋ + 1/2 s(s−1)n + i(13/2 i n + n/2 − ⌊(n−s)^2/4⌋) is at most s⌊(n−s)^2/4⌋ + s(s−1) + 13i^2 + i/2 n − i(n−s)^2/4 + i. This second inequality is false: it replaces the positive terms 1/2 s(s−1)n and 13/2 i^2 n by s(s−1) and 13i^2, respectively, losing a factor n. For the stated range n ≥ 30(s+1) and s ≥ 3, the term 1/2 s(s−1)n is already much larger than s(s−1), so the right-hand side is smaller, not larger, than the expression on the left. The subsequent definition of f(n,s,i) and the estimates f(n,s,s−2) < 0 and f(n,s,s/2) < 0 rely on this invalid bound. Because this is the only estimate in Case 2 that forces |H| below s⌊(n−s)^2/4⌋, the upper bound in Theorem 1.6 for s ≥ 3 is not established as written.","section":"Section 4, Case 2"},{"comment":"In the equality analysis of Case 1, the paper states that if a triangle xyz exists in G_i, then by (2.1) one obtains deg_{G_i}(x)+deg_{G_i}(y)+deg_{G_j}(x)+deg_{G_j}(y) ≤ 2n−2 for j ≠ i, and calls this 'a contradiction'. However, (2.1) only yields deg_{G_j}(x)+deg_{G_j}(y) ≤ n−2, and since xy is not known to be an edge of G_j, this does not conflict with the equality condition deg_{G_j}(u)+deg_{G_j}(v)=n for edges uv of G_j. No contradiction is apparent without an additional argument. The equality characterization in Theorem 1.8 is therefore not proven, and this gap propagates to the uniqueness part of Theorem 1.6. Please supply the missing derivation or revise the argument.","section":"Section 2, proof of Theorem 1.8"}],"minor_comments":[{"comment":"The proof refers to 'by Claim 4' when bounding deg(1, ¯S), but Claim 4 appears only inside Proposition 3.4; the claim should be restated or its location clarified in Section 4 to avoid confusion.","section":"Section 4, Case 1"},{"comment":"There are minor language issues: 'we showed' in the abstract is informal, and 'an 2-colored' in the abstract should be 'a 2-colored'; a careful proofread of the English would improve the presentation.","section":"Section 1 and Abstract"},{"comment":"In the same display as the major error, the coefficient of i n/2 appears to have the wrong sign relative to the expansion of binom(s−i,2)n; the algebraic manipulation should be revisited and corrected consistently, including the factor-n issue noted in the major comment.","section":"Section 4, Case 2"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a natural and timely question, and the proof strategy is well-motivated. However, the invalid algebraic inequality in Section 4, Case 2, is a genuine load-bearing error that invalidates the proof of the main upper bound as written. The result may well be true and the gap seems potentially fixable, but the manuscript is not yet ready for acceptance. I recommend major revision and a careful re-check of the algebraic transitions and the equality case of Theorem 1.8."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Three things to know up front. The main result is genuinely new: ex_3(n,{F5,M^3_{s+1}}) = binom(n−1,2) for s=1,2 and s floor((n−s)^2/4) for s≥3 when n≥30(s+1), with the expected unique extremal hypergraphs. The paper also proves a 2-colored Mantel theorem (Theorem 1.8) that looks correct and is a useful standalone lemma. But there is a load-bearing algebraic error in Case 2 of Theorem 1.6. As written, the chain from (4.1) to the definition of f(n,s,i) drops a factor of n in two positive terms: 1/2 s(s−1)n becomes s(s−1), and (13/2)i^2 n becomes 13i^2. The right-hand side is smaller than the left, so the asserted inequality is false. The subsequent convexity argument for f(n,s,i) rests on this step, so the upper bound for s≥3 is not established as written.\n\nThe good news: this looks like a typo rather than a conceptual failure. The surrounding structure is coherent. Lemma 3.3 is a solid degree-ordering tool, the reduction to link graphs and 2-colored triangle-free graphs is natural, and the s≤2 case goes through cleanly. The equality statement in Theorem 1.8 has a small omitted derivation—when a triangle in G_i contradicts (2.1), you need to invoke the equality condition deg_Gi(x)+deg_Gi(y)=n and deg_Gj(x)+deg_Gj(y)=n to get 2n, which is one line. That's minor.\n\nThe paper is worth refereeing. The result resolves a concrete open case in a recent research program, and the 2-colored Mantel theorem is a reusable tool. The gap is confined to one display and should be fixable; the authors need to supply a valid bound, probably by keeping the n-dependent terms and showing their maximum is negative under the n≥30(s+1) hypothesis. I would send it to a serious referee with a request for a corrected Case 2.","headline":"New result and a nice 2-colored Mantel lemma, but the main upper bound for s≥3 currently has a load-bearing algebraic gap in Case 2 of Theorem 1.6 that needs fixing.","tokens_in":16134,"tokens_out":3649,"would_cite":true,"duration_ms":33911,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C65","05D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper determines the exact Turán number of the generalized triangle $F_5$ among 3-graphs with bounded matching number, and proves a 2-colored Mantel theorem as the key tool.","keywords":["Turán number","generalized triangle","F5-free hypergraph","matching number","2-colored Mantel theorem","extremal hypergraph","link graph","hypergraph Turán problem"],"falsifier":"Substitute $s=3$, $n=120$, and $i=1$ into the displayed inequality in Case 2 of Theorem 1.6's proof: the left-hand side contains $\\tfrac{1}{2}s(s-1)n=360$ while the printed right-hand side uses $s(s-1)=6$, so the claimed inequality is false ($-2222.25$ is not at most $-2575.25$); this one substitution refutes the proof step, so the theorem's upper bound is not proved by the argument as written unless the step is corrected.","tokens_in":15048,"feed_emoji":"🧩","tokens_out":14045,"duration_ms":141495,"temperature":0.7,"pith_summary":"This paper determines the exact maximum number of edges in a 3-uniform hypergraph on $n$ vertices that contains no copy of the generalized triangle $F_5$ and no matching of size $s+1$. For $n \\geq 30(s+1)$, the maximum is $\\binom{n-1}{2}$ when $s=1,2$, attained only by the full star at one vertex, and $s\\lfloor (n-s)^2/4 \\rfloor$ when $s \\geq 3$, attained only by the complete 3-partite hypergraph $H_3(n,s)$ with part sizes $s$, $\\lfloor (n-s)/2 \\rfloor$, and $\\lceil (n-s)/2 \\rceil$. The proof rests on a new 2-colored version of Mantel's theorem: $p$ graphs on the same $n$ vertices with no triangle colored with two edges of one color and one of another have at most $p\\lfloor n^2/4 \\rfloor$ edges in total. This matters because $F_5$ has chromatic number 2 yet positive Turán density, so the bounded-matching problem is not covered by the known result for hypergraphs of chromatic number greater than 2.","feed_headline":"F5-free 3-graph edge maximum found under matching bound","feed_subtitle":"For large n the unique extremal hypergraph is the complete 3-partite hypergraph with one part of size s.","key_machinery":"The central objects are the generalized triangle $F_5$, the 3-graph on five vertices with edges $abc, abd, cde$, and the extremal candidate $H_3(n,s)$, the complete 3-partite 3-graph with part sizes $s$, $\\lfloor (n-s)/2 \\rfloor$, and $\\lceil (n-s)/2 \\rceil$. The main tool is the 2-colored Mantel theorem (Theorem 1.8), proved by induction on $n$ via a degree-sum inequality and an equality analysis. The $F_5$-free condition makes the link graphs $L(v,\\bar S)$ for $v$ in a selected $s$-set jointly 2-colored triangle-free, so Theorem 1.8 bounds their total contribution; its equality case forces all those links to be the same balanced complete bipartite graph, which is what singles out $H_3(n,s)$. Lemma 3.3 is the structural reduction that selects the set $S$ and decomposes the edge count into link contributions and small error terms.","core_discovery":"The central claim is Theorem 1.6: for $n \\geq 30(s+1)$, every $F_5$-free 3-graph with matching number at most $s$ has at most $\\binom{n-1}{2}$ edges when $s=1,2$ and at most $s\\lfloor (n-s)^2/4 \\rfloor$ edges when $s \\geq 3$, with equality only for the full star in the first case and only for $H_3(n,s)$ in the second. The engine is Theorem 1.8, the 2-colored Mantel theorem: if $G_1,\\ldots,G_p$ are graphs on a common $n$-set and no triangle uses two edges of one color and a third edge of another color, then $\\sum_i e(G_i) \\leq p\\lfloor n^2/4 \\rfloor$, and equality forces $G_1=\\cdots=G_p=K_{\\lfloor n/2 \\rfloor,\\lceil n/2 \\rceil}$. Applying this to the link graphs of vertices outside a carefully chosen $s$-set bounds the total edge count and, through the equality case, identifies the extremal hypergraph.","pith_inferences":["The same scheme—select an $s$-set, pass to link graphs, and apply a colored Mantel bound—should transfer to other 3-graphs whose forbidden configurations make link graphs colored-triangle-free; a natural test case is the analogue of $F_5$ for other 2-chromatic, positive-density hypergraphs.","The $n \\geq 30(s+1)$ hypothesis is probably larger than needed; the authors' Conjecture 5.1 asks for $n \\geq 3(s+1)$, and exhaustive search for small pairs such as $s=3$ and $n=12,\\ldots,30$ could test that range before a full proof.","A stability version of Theorem 1.8—near-equality forcing each graph to be close to the same balanced complete bipartite graph—would turn the uniqueness argument into a stability theorem for $H_3(n,s)$, with consequences for approximate counting of extremal hypergraphs."],"forward_implications":["For every admissible $n$ and $s \\geq 3$, the construction from the ordinary $F_5$ Turán problem, the complete 3-partite hypergraph $H_3(n,s)$, is the unique maximizer once the matching number is capped at $s$.","The 2-colored Mantel theorem extends the classical Mantel theorem: taking all $p$ graphs identical recovers Mantel's bound, so the new theorem is a genuine common generalization.","Corollary 1.9 is immediate: if $G_1,\\ldots,G_p$ are pairwise 2-colored triangle-free on the same $n$ vertices, then $\\prod_i e(G_i) \\leq \\lfloor n^2/4 \\rfloor^p$, with equality only in the balanced bipartite case.","The result adds $F_5$ to the small family of forbidden hypergraphs with chromatic number 2 and positive Turán density for which the bounded-matching Turán number is known exactly."],"supporting_citations":[{"why":"Supplies Lemma 1.2, which turns high vertex degrees into a large matching in the structural Lemma 3.3.","marker":"[1]"},{"why":"Supplies the Erdős–Gallai theorem used to bound the size of link subfamilies in the s=2 case.","marker":"[3]"},{"why":"Supplies the exact Erdős–Ko–Rado theorem used to handle intersecting cases for s=1 and s=2.","marker":"[4],[6],[17]"},{"why":"Determines the F5 Turán number and identifies the complete 3-partite construction that H3(n,s) extends.","marker":"[8]"},{"why":"Supplies the Hilton–Milner bound for non-star intersecting families used in the s=2 proof.","marker":"[11]"}],"fun_headline_variants":["Exact edge bound for F5-free 3-graphs with bounded matching","Sharp edge count for F5-free hypergraphs with matching number at most s","Bounding edges of F5-free 3-graphs via 2-colored Mantel"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire $s \\geq 3$ case depends on one algebraic inequality in Case 2 of the proof of Theorem 1.6; as printed, it replaces a term proportional to $n$ by a constant, and if that step cannot be repaired the upper bound is not established.","fun_headline_variants_meta":{"raw":{"variants":["Exact edge bound for F5-free 3-graphs with bounded matching","Sharp edge count for F5-free hypergraphs with matching number at most s","Bounding edges of F5-free 3-graphs via 2-colored Mantel"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00081,"raw_usage":{"total_tokens":3552,"prompt_tokens":940,"completion_tokens":2612,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":2543}},"tokens_in":556,"tokens_out":2612,"duration_ms":18983,"temperature":1.0,"reasoning_tokens":2543,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:50:55.661340+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Substitute $s=3$, $n=120$, and $i=1$ into the displayed inequality in Case 2 of Theorem 1.6's proof: the left-hand side contains $\\tfrac{1}{2}s(s-1)n=360$ while the printed right-hand side uses $s(s-1)=6$, so the claimed inequality is false ($-2222.25$ is not at most $-2575.25$); this one substitution refutes the proof step, so the theorem's upper bound is not proved by the argument as written unless the step is corrected.","supporting_citations":[{"cited_title":"Akiyama and P","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 1.2, which turns high vertex degrees into a large matching in the structural Lemma 3.3."},{"cited_title":"Erd˝ os and T","cited_arxiv_id":null,"evidence_quote":"Supplies the Erdős–Gallai theorem used to bound the size of link subfamilies in the s=2 case."},{"cited_title":"Frankl and Z","cited_arxiv_id":null,"evidence_quote":"Determines the F5 Turán number and identifies the complete 3-partite construction that H3(n,s) extends."},{"cited_title":"Hilton and E.C","cited_arxiv_id":null,"evidence_quote":"Supplies the Hilton–Milner bound for non-star intersecting families used in the s=2 proof."}],"review_version":1}