{"id":"337fa95b-1ee2-4608-bbdc-cc3b6251c1d3","arxiv_id":"2505.17981","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every k ≥ 3, a large k-uniform hypergraph with minimum positive codegree at least (k−1)/k n − (k−2) and no isolated vertices must contain a perfect matching, and this bound is best possible.","lead":"This math paper finds the exact minimum positive codegree needed to guarantee a perfect matching in large hypergraphs, for every uniform edge size k. It closes the gap left by earlier work that was off by a constant amount.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.4's first case has a counting gap: a B-pair entirely inside a (k-1)-set S can trigger the delta+ drop without making any vertex of S incident to many B-pairs.","rationale":"I read the paper in good faith and found the main structure sound: Lemma 2.2 (extremal case), Lemma 3.1 (absorbing), and the Farkas-based Lemma 4.3 are largely correct, with the terse dominance step in Lemma 4.3 being plausible and verifiable by coordinatewise comparison. The reader's weakest assumption was the external Theorem 4.5, which I do not press: if its statement is accepted, it is the right Pippenger-Spencer-type tool. However, the proof of Corollary 4.4, which produces the small pair-weight fractional matching needed for Theorem 4.5, contains a genuine gap. The first case claims that many deleted edges containing a (k-1)-set S force some vertex of S to lie in epsilon n distinct B-pairs, but a single B-pair inside S accounts for all such edges in the k=3 example. This is a non sequitur in the written proof. The conclusion of Corollary 4.4 may still be true, and the gap may be repairable by an extremal argument, but the manuscript does not provide it. Because the non-extremal case of Theorem 1.2 depends on this corollary, the proof is not complete as written. I therefore recommend a conditional acceptance: the authors should repair or justify the counting step in the first case of Corollary 4.4 before the main theorem can be considered fully proved.","tokens_in":960,"tokens_out":1544,"duration_ms":407148,"concrete_test":"Formalize the disputed counting step with k=3, S={a,b}, B={{a,b}}, and let H contain all edges S union {v} for v in V\\S, with H'=H minus all edges containing {a,b}. Check whether the hypotheses of the first case of Corollary 4.4 are satisfied and whether any vertex of S lies in at least epsilon n pairs of B; the stated conclusion fails. To settle the mathematical claim, either supply the missing extremal-case argument showing H is gamma-extremal whenever the only heavy B-pairs lie inside S, or search small n for a high-delta+, non-gamma-extremal H whose minimizing fractional matching has B={a,b}; if such an H exists, Corollary 4.4 is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The weakest point is not Theorem 4.5 itself but the proof of Corollary 4.4, which is the only route to the small pair-weight fractional matching used by Lemma 4.1 and Theorem 4.5. In the first case of the proof, the paper argues: if delta+(H') < (k-1)/k n - (alpha+(k-1)epsilon)n, some (k-1)-set S is contained in more than (k-1)epsilon n deleted edges; each such edge contains a pair in B, at least one vertex of which must be in S, so 'some vertex u in S is in at least epsilon n pairs in B.' This last inference is not valid. For example, when k=3, take S={a,b} and B={{a,b}}. Every edge S union {v} is deleted because it contains the B-pair {a,b}, and neither a nor b is contained in more than one pair of B, even though there are Omega(n) deleted edges containing S. Thus the premise can hold while the stated conclusion fails. The proof would need an additional argument handling the configuration where all deleted edges through S are explained by a single B-pair inside S, but no such argument is given. Since Corollary 4.4 feeds directly into Lemma 4.1 and then Theorem 1.2, this gap is load-bearing for the non-extremal case.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper determines, for every k≥3, the exact minimum positive codegree threshold that forces a perfect matching in a large k-uniform hypergraph with no isolated vertices. Theorem 1.2 states that δ^+(H) ≥ (k−1)n/k − (k−2) suffices for all sufficiently large n divisible by k, and the construction of Halfpap and Magnan shows this is best possible. The proof splits into an extremal case handled by deleting a small matching and applying the Daykin–Häggkvist theorem, an absorbing lemma built from many small absorbing sets, and a non-extremal case that passes through perfect fractional matchings with small pair weights and then invokes a Pippenger–Spencer-type theorem.","tokens_in":11461,"tokens_out":21548,"duration_ms":209402,"significance":"If correct, the result is a clean and complete resolution of the positive-codegree threshold problem for perfect matchings, improving on the k=3 result of Halfpap and Magnan and on their bounds for k≥4. The proof is remarkably short for this type of exact threshold and combines classical tools (Daykin–Häggkvist, Farkas's lemma, Pippenger–Spencer) with a carefully tailored absorbing argument. The extremal construction showing sharpness is simple and correct. The main reservation is that the non-extremal case leans on Theorem 4.5, restated from the recent preprint [2] by the first author's group; the present manuscript does not prove that input, so the reader cannot fully verify that step from this paper alone.","major_comments":[{"comment":"The inference 'each such edge contains a pair in B, at least one vertex of which must be in S. We conclude that some vertex u∈S is in at least εn pairs in B' is not valid as stated: a single B-pair contained entirely in S can lie in all edges S∪{x} and thus account for Ω(n) deleted edges without making any vertex of S incident to many B-pairs. The gap is repairable, because a set S witnessing the δ^+ drop has deg_{H'}(S)≥1, and if a B-pair were contained in S then every edge containing S would be deleted, forcing deg_{H'}(S)=0. Once that observation is added, every deleted edge containing S must contain a B-pair of the form {s,x} with s∈S and x∉S, and each such pair contributes at most one edge S∪{x}, so the stated bound on some u∈S follows. This missing argument should be supplied, since Corollary 4.4 feeds directly into Lemma 4.1 and hence Theorem 1.2.","section":"Corollary 4.4, first case (δ^+ drop)"},{"comment":"The line '1 = ∑_{e∈E_u(H)} w(e) ≥ M εn' is false: an edge containing u may contain several vertices v with uv∈B, so the sum ∑_{v:uv∈B} w_{uv} can be as large as (k−1)∑_{e∋u}w(e) = k−1, not at most 1. The correct conclusion from u being in at least εn pairs of B is k−1 ≥ M εn, i.e. M ≤ (k−1)/(εn). Since the later application only needs pair weights o(1), this can be repaired by choosing ε smaller (for example replacing ε by ε/(k−1) in the corollary), but the displayed inequality as written is not justified.","section":"Corollary 4.4, final paragraph"},{"comment":"The assertion deg_H(v) ≥ (n/2)^{k−1} is not a consequence of the assumed minimum positive codegree condition. Proposition 1.3(ii) gives deg_H(v) ≥ binom(δ^+(H)+k−2, k−1), which is asymptotically (((k−1)/k)^{k−1}/(k−1)!) n^{k−1}, a positive constant times n^{k−1} but generally smaller than (n/2)^{k−1}. The desired conclusion deg_H(v) ≥ 2ε n^{k−1} still follows for ε sufficiently small, so the error is local, but the displayed inequality should be corrected or replaced by a bound with the correct constant.","section":"Corollary 4.4, isolated-vertex case"}],"minor_comments":[{"comment":"The assertion 'from which it follows that y·x_e ≥ y·x_{B_i}' is terse; adding one sentence explaining that the index multiset of e majorizes that of B_i (and likewise for e' and A_i) would improve readability.","section":"Lemma 4.3"},{"comment":"The statement that the number of options for each choice of b_i is '> n/4' is asserted quickly; a short derivation showing δ^+(H)−(n−|B|) > n/4 for the relevant range would be helpful.","section":"Lemma 2.2"},{"comment":"There are a few minor typographical issues, including inconsistent spacing in expressions such as 'deg( S)' and the use of 'H' instead of 'Hext' in the sentence describing the extremal construction; these should be cleaned up.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central proof is essentially sound and the identified problems in Corollary 4.4 are locally repairable, so I do not see grounds for rejection. The main risk is the dependence on Theorem 4.5, which is quoted from the unpublished preprint [2]; I would encourage the editor to ask the authors either to include a self-contained proof of that statement or to confirm that [2] has been independently verified. The paper is otherwise a strong and significant contribution to extremal hypergraph theory."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves the exact positive codegree threshold for perfect matchings in k-uniform hypergraphs for all k ≥ 3, sharpening Halfpap–Magnan's additive-constant result. The extremal case and the absorbing lemma are clean, modular, and check out. The construction showing optimality is standard and correct. If the proof were complete, this would be a solid within-field contribution.\n\nThe problem is in the non-extremal case, specifically Corollary 4.4. In the first case, the authors argue that if some (k−1)-set S is contained in more than (k−1)εn deleted edges, then each such edge contains a pair from B, so at least one vertex of S is in at least εn pairs of B. That inference is invalid. The deleted edges could all contain the same pair from B that lies entirely inside S. For k = 3, take S = {a,b} and B = {{a,b}}; then every edge {a,b,v} is deleted, but neither a nor b is in more than one pair of B. The number of edges through S can be linear while the number of B-pairs incident to S stays constant. The proof needs an additional argument to handle this configuration, and none is given. Since Corollary 4.4 is the only route to the small pair-weight fractional matching used in Lemma 4.1 and Theorem 4.5, this gap is load-bearing.\n\nA lesser concern is Theorem 4.5, restated from an unreviewed preprint by the same group. That restatement is of a known Pippenger–Spencer-type result, so it is not a red flag, but it does mean the paper's main theorem depends on a recent, not-yet-vetted source.\n\nOverall, the theorem is probably true and the framework is sound. The gap in Corollary 4.4 looks repairable, but it is not a minor typo. I would send this to a serious referee and ask for a full fix of that counting argument before acceptance.\n\nFor the reading group: yes, it is worth discussing precisely because the flaw is instructive. I would not cite it as a proven result until the gap is closed.","headline":"Likely-true result with a real gap in the counting argument of Corollary 4.4; worth reviewing, but not acceptable as-is.","tokens_in":11999,"tokens_out":4516,"would_cite":false,"duration_ms":36437,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","05C70","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every $k\\ge3$, the exact minimum positive codegree forcing a perfect matching in large $k$-uniform hypergraphs is $\\frac{k-1}{k}n-(k-2)$, and the bound is sharp.","keywords":["positive codegree","perfect matching","k-uniform hypergraphs","extremal threshold","fractional matching","absorbing method","Dirac-type problem","sharpness construction"],"falsifier":"Compute the minimum positive codegree of the split-graph construction with $|A|=n/k+1$ and edges all $k$-sets with $|e\\cap A|\\le1$: it equals $\\frac{k-1}{k}n-(k-1)$, one below the theorem's threshold, and it has no perfect matching. If any $k$-graph with no isolated vertices, $\\delta^+(H)\\ge\\frac{k-1}{k}n-(k-2)$, and no perfect matching could be found for large $n$, Theorem 1.2 would be false; the sharpness example shows exactly how close one can get without crossing the theorem's line.","tokens_in":10965,"feed_emoji":"🧩","tokens_out":15386,"duration_ms":107726,"temperature":0.7,"pith_summary":"This paper settles, for every uniformity $k\\ge 3$, the exact minimum positive codegree that forces a $k$-uniform hypergraph on $n$ vertices to contain a perfect matching. The theorem states that if $n$ is sufficiently large and divisible by $k$, and $H$ has minimum positive codegree $\\delta^+(H)\\ge \\frac{k-1}{k}n-(k-2)$ and no isolated vertices, then $H$ has a perfect matching. A construction shows the hypothesis cannot be weakened: for every $n$ divisible by $k$ there is a $k$-graph with $\\delta^+(H)=\\frac{k-1}{k}n-(k-1)$, no isolated vertices, and no perfect matching. Earlier work had the exact threshold only for $k=3$ and, for $k\\ge4$, had bounds tight only up to an additive constant; here the constant is removed for all $k$. The consequence is that a much weaker degree notion still forces the spanning matching once isolated vertices are excluded.","feed_headline":"Exact positive codegree threshold for hypergraph perfect matchings","feed_subtitle":"Minimum positive codegree (k-1)n/k -(k-2), with no isolated vertices, forces a perfect matching; the bound is sharp.","key_machinery":"The argument combines three mechanisms. First, a random selection of small sets that can absorb any leftover $k$-set produces an absorbing set $A$ of size $O(\\beta n)$; whenever the rest of the proof leaves at most $\\beta^2 n$ uncovered vertices, $A$ together with those vertices admits a perfect matching. Second, in the extremal case a small matching is deleted to cover all atypical vertices, the leftover vertices are partitioned into $k$ equal classes, and a classical $k$-partite hypergraph matching theorem supplies a perfect matching of the remainder. Third, in the non-extremal case linear-programming duality proves the existence of a perfect fractional matching, refines it so every pair of vertices carries small total weight, and a recent approximate matching theorem converts it into an integral matching covering all but $\\eta n/k$ vertices. The conversion theorem is the step that lets fractional evidence become a genuine near-perfect matching and is therefore the main load-bearing input of the non-extremal proof.","core_discovery":"The central claim is Theorem 1.2: for every $k\\ge3$ there is an $n_0$ such that, whenever $k\\mid n$ and $n\\ge n_0$, every $k$-uniform hypergraph $H$ on $n$ vertices with $\\delta^+(H)\\ge \\frac{k-1}{k}n-(k-2)$ and no isolated vertices contains a perfect matching. The constant is best possible. The sharpness witness is the construction in which the vertex set is split into $A$ of size $n/k+1$ and $B$ of the remaining vertices, with edges exactly the $k$-sets meeting $A$ in at most one vertex; its minimum positive codegree is $\\frac{k-1}{k}n-(k-1)$, it has no isolated vertices, yet every matching uses at most one vertex of $A$ per edge and so cannot cover $A$. The proof separates graphs that are close to this construction from graphs that are far from it, and handles the two regimes with different matching machinery.","pith_inferences":["Beyond the paper, a likely extension is the exact positive codegree threshold for tight Hamilton cycles, since the extremal construction here is exactly the obstruction used for cycle problems; the non-extremal conversion theorem would be the crucial input to upgrade.","Beyond the paper, the proof's use of the no-isolated-vertices condition is indirect, so the same argument may adapt to hypergraphs with a bounded number of isolated vertices, with an additive shift in the threshold.","Beyond the paper, the additive constant $-(k-2)$ arises from counting extensions of $(k-1)$-sets; computational search for small $k$ could test whether the threshold remains exact with a much smaller $n_0$ than the proof supplies."],"forward_implications":["For each $k\\ge3$, the exact threshold for a large perfect matching is $\\delta^+(H)\\ge\\frac{k-1}{k}n-(k-2)$ under the no-isolated-vertices assumption, and the threshold is sharp.","The $k=3$ case and the $k\\ge4$ near-results are subsumed, with the additive ambiguity removed in every uniformity.","The announced result that the same degree condition forces a tight Hamilton cycle would imply this theorem, since a tight cycle contains a perfect matching; the present proof is a shorter, matching-specific argument.","Any hypergraph meeting the threshold is either visibly close to the extremal construction, where a direct $k$-partite argument applies, or admits both an absorbing set and a near-perfect matching, so the two regimes together cover all cases."],"supporting_citations":[{"why":"Established the $k=3$ case exactly, gave bounds for $k\\ge4$ tight up to an additive constant, and supplied the construction showing sharpness.","marker":"[5]"},{"why":"Supplies the classical $k$-partite hypergraph matching theorem used in the extremal case after a small matching is deleted.","marker":"[3]"},{"why":"Provides the conversion theorem from fractional matchings to an almost-perfect integral matching; this is the load-bearing non-extremal step.","marker":"[2]"},{"why":"Underlies the approximate matching theorem used in the conversion, guaranteeing large matchings in approximately regular hypergraphs.","marker":"[13]"},{"why":"Gives the exact minimum codegree threshold for perfect matchings, the classical benchmark for this family of problems.","marker":"[15]"},{"why":"Introduced minimum positive codegree, the degree notion whose exact threshold this paper determines.","marker":"[1]"},{"why":"Supplies the sharpened edge-extension proposition used throughout to build edges one vertex at a time.","marker":"[12]"},{"why":"Gives the binomial concentration inequality used in the random construction of the absorbing set.","marker":"[8]"}],"fun_headline_variants":["Sharp codegree bound forces perfect matchings","Exact threshold: positive codegree for hypergraph matchings","Optimal positive codegree condition for perfect matchings","Hypergraph matchings: exact positive codegree needed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The non-extremal case rests on a recently stated theorem that any fractional matching with each vertex carrying weight at least $1-\\varepsilon$ and each pair of vertices sharing total weight at most $\\varepsilon$ can be converted into an ordinary matching covering all but a small fraction of the vertices; if that conversion is wrong at the stated precision, the main theorem does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sharp codegree bound forces perfect matchings","Exact threshold: positive codegree for hypergraph matchings","Optimal positive codegree condition for perfect matchings","Hypergraph matchings: exact positive codegree needed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000446,"raw_usage":{"total_tokens":2221,"prompt_tokens":883,"completion_tokens":1338,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":1275}},"tokens_in":499,"tokens_out":1338,"duration_ms":10839,"temperature":1.0,"reasoning_tokens":1275,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:37:34.460105+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the minimum positive codegree of the split-graph construction with $|A|=n/k+1$ and edges all $k$-sets with $|e\\cap A|\\le1$: it equals $\\frac{k-1}{k}n-(k-1)$, one below the theorem's threshold, and it has no perfect matching. If any $k$-graph with no isolated vertices, $\\delta^+(H)\\ge\\frac{k-1}{k}n-(k-2)$, and no perfect matching could be found for large $n$, Theorem 1.2 would be false; the sharpness example shows exactly how close one can get without crossing the theorem's line.","supporting_citations":[{"cited_title":"Positive co-degree thresholds for spanning structures","cited_arxiv_id":"2409.09185","evidence_quote":"Established the $k=3$ case exactly, gave bounds for $k\\ge4$ tight up to an additive constant, and supplied the construction showing sharpness."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical $k$-partite hypergraph matching theorem used in the extremal case after a small matching is deleted."},{"cited_title":"Pippenger and J","cited_arxiv_id":null,"evidence_quote":"Underlies the approximate matching theorem used in the conversion, guaranteeing large matchings in approximately regular hypergraphs."},{"cited_title":"R¨ odl, A","cited_arxiv_id":null,"evidence_quote":"Gives the exact minimum codegree threshold for perfect matchings, the classical benchmark for this family of problems."},{"cited_title":"Balogh, N","cited_arxiv_id":null,"evidence_quote":"Introduced minimum positive codegree, the degree notion whose exact threshold this paper determines."},{"cited_title":"Positive codegree thresholds for Hamilton cycles in hypergraphs","cited_arxiv_id":"2505.11400","evidence_quote":"Supplies the sharpened edge-extension proposition used throughout to build edges one vertex at a time."},{"cited_title":"Janson, T","cited_arxiv_id":null,"evidence_quote":"Gives the binomial concentration inequality used in the random construction of the absorbing set."}],"review_version":1}