{"id":"db056b63-4f5a-4ddf-b118-b2bb15804180","arxiv_id":"2608.04615","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For large vertex sets, the paper determines the exact maximum of the (t,p)-norm and its unique extremal hypergraph in three classical settings: bounded matching number, k-intersecting families, and hypergraphs avoiding linear paths.","lead":"The paper finds the exact largest value of a degree-power score on set families when matchings, intersections, or linear paths are forbidden. It gives the unique extremal family in each case once the vertex set is large, unifying and extending several classical extremal results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Path theorems 1.5–1.6 rest on unproved cleaning Lemmas 5.6–5.7; if Lemma 5.7's exact 2-intersecting conclusion fails, the even-path uniqueness collapses.","rationale":"I read the main arguments in Sections 3 and 4 as internally coherent. The shifting proof of Theorem 1.1 has a few compressed steps, such as shifting the s disjoint edges into [2, sr+1], but these are standard and do not appear to conceal an error. The degree-pair majorization in Lemma 3.1 is valid, the induction maintains the background parameter uniformly, and the concave case correctly uses subadditivity plus the non-star stability lemma. Similarly, the EKR proof in Section 4 is clean: the non-trivial family has size O(n^{r-k-1}), and the uniform background estimates in Lemma 4.1 make the comparison with the full k-star work for every c ≥ 0. Theorem 5.8 is the strongest part of the paper and appears sound; the high-shadow, weighted-pair, and weighted-vertex arguments each reduce to the cited edge stability theorem of Kostochka–Mubayi–Verstraëte after carefully controlling the lower-order degree contributions. The exact path proofs, however, depend on the cleaning lemmas 5.6 and 5.7, which are stated without proof and are not standard textbook results. Lemma 5.7 is particularly delicate: the norm comparison requires B0 to be exactly 2-intersecting, and the uniqueness of the extremal family E_{n,r,s}(A,Q) relies on applying Theorem 1.3 with k = 2 to B0. If the lemma only yields almost-2-intersection, or if the cleaning constant α depends on the background parameter in a way not covered by the statement, the conclusion 'M = ∅, B = B0' is not justified. The reader identified the same weakest point, and I agree; this is not a demonstrated error but a load-bearing external dependency. A careful reconstruction of Lemma 5.7 from [20, Section 6.4] would settle whether the paper's path theorems are fully established. Until then, conditional acceptance is the appropriate verdict.","tokens_in":17699,"tokens_out":48991,"duration_ms":531896,"concrete_test":"Recover the proof of Lemma 5.7 from Kostochka–Mubayi–Verstraëte [20, Section 6.4] and check that their cleaning step produces a fully 2-intersecting B0 (not merely almost 2-intersecting), with |B\\B0| ≤ η|M| for every near-extremal P^r_{2s}-free H satisfying the edit-distance hypothesis, and that H_{n,r,A} ∪ B0 is P^r_{2s}-free. If the published proof yields only an almost-2-intersecting B0, or requires H to be edge-extremal in a stronger sense than |H△H_{n,r,A}| = o(n^{r-1}), then the step 'extremality forces M = ∅, B = B0' in Theorem 1.6 does not follow and the even-path theorem is unproved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The exact path theorems are the only place where the paper relies on structural statements that are asserted but not proved. Lemma 5.7 is the most load-bearing: it claims that any P^r_{2s}-free H within α n^{r-1} of the full (s−1)-star admits a 2-intersecting subfamily B0 ⊆ B with |B\\B0| ≤ η|M| and with H_{n,r,A} ∪ B0 still P^r_{2s}-free. The subsequent norm comparison in §5.3.2 uses the full strength of this conclusion: extremality forces M = ∅ and B = B0, and then Theorem 1.3 with k = 2 requires B0 to be exactly 2-intersecting. A lemma that only gave an almost-2-intersecting cleaning, or that required edge-extremality in a stronger sense rather than the edit-distance o(n^{r-1}) supplied by Theorem 5.8, would break the uniqueness argument. The paper verifies only P-freeness, the norm lower bound, and |B|+|M| = o(n^{r-1}) before invoking these lemmas; it does not reconstruct their proof from [20, Section 6.4] or give a precise lemma/page reference. Since Theorems 1.5 and 1.6 depend on these lemmas, the central path characterization is conditional on the correctness and exact form of external statements.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the (t,p)-norm of r-uniform hypergraphs, defined as the sum of the p-th powers of all t-subset degrees, and determines its maximum over three classical extremal classes for sufficiently large n. Theorems 1.1 and 1.2 treat r-graphs with matching number at most s in the convex range p>1 and the concave range 0<p<1, respectively, with the full s-star H_{n,r,s} as the unique extremal family. Theorems 1.3 and 1.4 establish the analogous Erdős–Ko–Rado-type results for k-intersecting families, with the full k-star as the unique maximizer. Theorems 1.5 and 1.6 treat P^r_ℓ-free families in the convex range p>1: for odd ℓ=2a+1 the full a-star is extremal and unique, while for even ℓ=2s the extremal family is the star-plus-2-star construction E_{n,r,s}(A,Q). The proofs use shifting, coordinatewise degree comparisons, weighted high-degree and high-pair estimates, and stability-to-exact reductions.","tokens_in":17932,"tokens_out":31602,"duration_ms":331564,"significance":"If the quoted structural lemmas are valid, the paper fully resolves the (t,p)-norm versions of three foundational extremal problems, including uniqueness of extremal families, and covers the full range 1≤t≤r−1 rather than only t=r−1. Theorems 1.1–1.4 are self-contained and the proofs are structured around explicit, parameter-free comparisons: shifting combined with Karamata's inequality, Frankl's matching theorem and the EKR stability bound. The path results in Theorems 1.5–1.6 successfully connect the asymptotic norm problem to KMV stability through weighted high-shadow arguments, and the exact formulas (1)–(2) are concrete and checkable. The main caveat is that the even- and odd-path exact steps depend on two cleaning lemmas quoted from [20, Section 6.4] that are not proved in this manuscript; because the uniqueness statement for P^r_{2s}-free families uses the full strength of Lemma 5.7, the path theorems are conditional on those external statements being available in exactly the stated form.","major_comments":[{"comment":"The proofs of Theorems 1.5 and 1.6 depend on Lemma 5.6 and especially Lemma 5.7, which are stated without proof and referenced only to 'Kostochka–Mubayi–Verstraëte [20, Section 6.4]'. In the proof of Theorem 1.6, Lemma 5.7 is used in full strength: it is what forces M=∅ and B=B0, after which Theorem 1.3 with k=2 is applied to B0 and forces the outside family to be a full 2-star. If the exact 2-intersecting conclusion, or the assertion that H_{n,r,A}∪B0 remains P^r_{2s}-free, were to fail in the stated form, the uniqueness characterization E_{n,r,s}(A,Q) would collapse. The manuscript verifies only P-freeness, the norm lower bound, and |B|+|M|=o(n^{r-1}) before invoking Lemma 5.7, and it does not reconstruct the proof from [20] or give a precise lemma or page number. Please provide a proof of Lemmas 5.6 and 5.7, or quote the exact corresponding statement from [20], and confirm explicitly that the hypotheses of that statement are met by the norm-extremal families considered here.","section":"Section 5.3, Lemmas 5.6–5.7"}],"minor_comments":[{"comment":"The sentence 'We separate the cases p>1 and 0<p<1 because they has different proofs' contains a grammatical error: 'they has' should be 'they have'.","section":"Section 1.1"},{"comment":"In the concluding remarks, 'For seek of simplicity' should read 'For the sake of simplicity'.","section":"Section 6"},{"comment":"There are spacing errors of the form 'Whent=r−1' and 'anr-graph'; these should be 'When t=r−1' and 'an r-graph'.","section":"Abstract and throughout"},{"comment":"The threshold is written as N+(r,s,t), although p>1 is a parameter of the theorem; the proof suggests the threshold may be chosen independent of p, but this should be stated explicitly, or p should be included in the notation for clarity.","section":"Theorem 1.1"},{"comment":"Even if the lemmas are accepted as external results, a precise citation with theorem or lemma numbers from [20] would be much more useful than a section-level reference, given that the exact form of Lemma 5.7 is load-bearing for Theorem 1.6.","section":"Lemmas 5.6–5.7"},{"comment":"In the even-path proof, the sentence 'The contribution of t-sets meeting A is already fixed and equal to that of the full A-star' is correct, but it would help to add one clarifying sentence noting that this is because all outside edges are disjoint from A and hence contain no t-set meeting A.","section":"Section 5.3.2"}],"recommendation":"major_revision","confidential_remarks":"The core results for matching number and intersecting families (Theorems 1.1–1.4) appear sound and self-contained. The path theorems are credible but rest on the two cleaning lemmas quoted from [20, Section 6.4]; since Lemma 5.7's exact 2-intersecting conclusion is essential for the uniqueness statement of Theorem 1.6, I would like to see either a proof of that lemma or a precise quotation with page/lemma number before accepting. This is a fixable gap and not a fundamental error, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper delivers what it claims: exact maximizers and uniqueness for the (t,p)-norm in three classical settings—bounded matching number, k-intersecting families, and linear paths—for n large. The convex/concave split is handled with clean shifting and Jensen arguments, and the background parameter trick for p>1 is a nice touch. The path results for t=1 and t=2 are the genuinely new part; instead of importing stability of auxiliary rich graphs, they use weighted high-vertex and high-pair estimates that check out. I read the main estimates carefully and did not find a fatal gap.\n\nThe real soft spot is the even-path theorem. The uniqueness H ≅ E_n,r,s(A,Q) rests on Lemma 5.7, quoted from [20, Section 6.4], which asserts that in a near-extremal P^r_{2s}-free family the outside part can be cleaned to a 2-intersecting B_0 with tiny loss, and that the star plus B_0 remains P^r_{2s}-free. The paper checks only the norm lower bound and o(n^{r-1}) distance to the star before invoking it, and does not reconstruct the proof or give a precise lemma/page reference. If that lemma's exact conclusion fails, or its hypotheses don't match the norm-extremal family, the characterization collapses. That's a conditionality, not a demonstrated error, but a referee needs to verify it. The odd-path Lemma 5.6 is a similar but less risky import.\n\nThe novelty boundary with the recent sunflower/matching-norm preprints [24,25] is not drawn theorem-by-theorem. The paper cites them but doesn't say precisely which of its matching results overlap with their claims. That should be clarified, but it's not a correctness issue.\n\nNo circularity, no fitted constants, no target results assumed. The AI-use disclosure is explicit and doesn't affect the math. I'd send this to a serious referee. It deserves referee time; the path part especially needs a check of the imported lemmas.","headline":"Solid exact results for (t,p)-norm extremal problems, with the even-path uniqueness hinging on an unproved cleaning lemma that a referee should verify.","tokens_in":18517,"tokens_out":2268,"would_cite":true,"duration_ms":25735,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05D05","05C35","05C65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Star-type families uniquely maximize the $(t,p)$-norm under matching, intersection, and long-path restrictions.","keywords":["(t,p)-norm","hypergraph Turán problem","matching number","k-intersecting families","linear paths","stability","degree powers"],"falsifier":"Construct, for some $r\\ge 3$, $s\\ge 2$, $1\\le t\\le r-1$, and $p>1$, a $P^r_{2s}$-free $r$-graph on $n$ vertices with $\\|H\\|_{t,p} > \\|E_{n,r,s}\\|_{t,p}$ for arbitrarily large $n$; a concrete starting point is a full $(s-1)$-star with outside edges whose pairwise intersections contain two disjoint pairs, which would violate the $2$-intersecting cleaning conclusion. For the matching-number claims, a counterexample would be a family with matching number at most $s$ that is not contained in any $s$-star yet has norm exceeding $\\|H_{n,r,s}\\|_{t,p}$.","tokens_in":17451,"feed_emoji":"⭐","tokens_out":8829,"duration_ms":86776,"temperature":0.7,"pith_summary":"This paper proves that degree-power objectives select the same extremal families as ordinary edge-counting in three classical hypergraph problems. For $r$-graphs with matching number at most $s$ and for $k$-intersecting families, it shows that the natural star families uniquely maximize the $(t,p)$-norm for every $p>0$, provided the ground set is large enough; the convex case $p>1$ even holds with a constant background added to every degree. For hypergraphs with no long linear path, it proves the analogous statement for $p>1$, with the star being the unique maximizer for odd path length and a star plus a $2$-intersecting tail for even path length. The paper also gives the exact asymptotic value of the path extremal norm and characterizes all equality cases. A careful reader should note that the exact path results rely on two cleaning lemmas quoted from earlier path-stability work, which are not proved here.","feed_headline":"Star-type families maximize (t,p)-norm in three extremal problems","feed_subtitle":"For large n, degree-power sums are maximized only by star families, in both convex and concave ranges.","key_machinery":"The proof's main engine is a shifting operation that does not decrease the $(t,p)$-norm when $p\\ge 1$: replacing a larger vertex by a smaller one in an edge makes the pair of $t$-set degrees for the two vertices majorize the original pair, and a standard majorization inequality for convex functions turns this into a norm inequality. This lets the convex matching-number proof induct on $s$ by first forcing a full star at vertex $1$. In the concave range $0<p<1$, subadditivity $(x+y)^p\\le x^p+y^p$ and a power-mean bound replace majorization, and a stability lemma shows any non-star family loses a factor $(s-1)^p$ versus $s^p$. For path-free families, the argument splits by $t$: for $t\\ge 3$ a high $t$-shadow is itself a linear $t$-uniform path, so the edge-stability theorem applies to the shadow; for $t=2$ and $t=1$, weighted high-pair and high-vertex estimates control the norm, yielding the asymptotic $(a+o(1))\\binom{n}{t-1}D_t^p$ with $D_t=\\binom{n-t}{r-t}$. The exact step then compares gains and losses against the full star, using two cleaning lemmas (5.6 and 5.7) to reduce a near-extremal family to a star plus a small $2$-intersecting remainder.","core_discovery":"The central claim is that the $(t,p)$-norm, the sum over $t$-subsets of the $p$-th power of their degree, inherits the extremal stars of classical extremal set theory, with uniqueness of the extremal family. Theorems 1.1 and 1.2 show that among $r$-graphs on $[n]$ with matching number at most $s$, the family $H_{n,r,s}$ of all edges meeting a fixed $s$-set uniquely maximizes the norm for every $p>0$, for all sufficiently large $n$. Theorems 1.3 and 1.4 show the same for $k$-intersecting families, where the full $k$-star is the unique maximizer. Theorems 1.5 and 1.6 extend this to $P^r_\\ell$-free hypergraphs: for odd length $\\ell=2a+1$ the full $a$-star uniquely maximizes, and for even length $\\ell=2s$ the family $E_{n,r,s}(A,Q)$, consisting of all edges meeting a fixed $(s-1)$-set together with all edges outside it containing a fixed pair $Q$, is the unique maximizer, for $p>1$ and all $1\\le t\\le r-1$.","pith_inferences":["The same shifting-plus-majorization mechanism should transfer to any degree-type extremal problem whose ordinary extremal family is a full star, provided the objective is an increasing convex function of the degrees; the paper's opening discussion of sunflowers suggests such a transfer.","The concave range $0<p<1$ for path-free families is left open; because concavity favors spread-out degree mass, the extremal family there need not be the star, so the restriction $p>1$ may be essential rather than technical.","Theorem 1.1's uniformity in $c$ implies a stronger statement: the star maximizes not just the $(t,p)$-norm but every functional obtained by integrating an increasing convex function of the degree vector.","A natural test of the even-path result would be to check numerically, for small $r,t,p$, whether any near-extremal $P^r_{2s}$-free family can violate the cleaning lemma's conclusion; the paper verifies only the $o(n^{r-1})$ distance before invoking it."],"forward_implications":["For bounded matching number, the inequality holds with an arbitrary background $c$ added to every degree when $p>1$, so the star wins even when the objective is a shifted power sum.","For $0<p<1$, concavity changes the proof but not the answer: the same star remains the unique maximizer in the matching and intersecting settings.","For $k$-intersecting families, the full $k$-star is the unique norm maximizer, giving a degree-power version of the classical intersection theorem.","For $P^r_\\ell$-free hypergraphs, the extremal family is the $a$-star for odd $\\ell=2a+1$ and the star-with-tail family $E_{n,r,s}(A,Q)$ for even $\\ell=2s$, with the asymptotic maximum $(a+o(1))\\binom{n}{t-1}D_t^p$.","All equality statements are exact isomorphisms, not just asymptotic: for large $n$, any family attaining the norm is the stated star or star-with-tail family."],"supporting_citations":[{"why":"Supplies the complete nontrivial-intersection theorem used to bound the norm of non-star $k$-intersecting families in Lemma 4.1.","marker":"[1]"},{"why":"Gives the recent degree-power results for intersecting families that the $k$-star theorems extend and sharpen.","marker":"[7]"},{"why":"Supplies the classical intersection theorem and the non-star size bound used throughout Section 4.","marker":"[11]"},{"why":"Supplies the exact edge-extremal bound for bounded matching number used as the base case and for equality in Theorems 1.1 and 1.2.","marker":"[14]"},{"why":"Provides the path Turán asymptotics, edge stability, and the two cleaning lemmas (5.6 and 5.7) on which Theorems 1.5 and 1.6 rest.","marker":"[20]"}],"fun_headline_variants":["Star-type families maximize (t,p)-norm in three settings","(t,p)-norm extremals: star-type families win uniquely","For large n, degree-power sums pick star-type families","Three extremal problems, one maximizer: star-type families"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exact path characterizations rest on two cleaning lemmas quoted from earlier work (Lemmas 5.6 and 5.7) that are not proved in this paper; the even-length lemma in particular asserts that a near-extremal path-free family can be reduced to a star plus a small $2$-intersecting remainder without creating the forbidden path, and if that assertion fails, the identification of $E_{n,r,s}(A,Q)$ as the unique maximizer collapses.","fun_headline_variants_meta":{"raw":{"variants":["Star-type families maximize (t,p)-norm in three settings","(t,p)-norm extremals: star-type families win uniquely","For large n, degree-power sums pick star-type families","Three extremal problems, one maximizer: star-type families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000466,"raw_usage":{"total_tokens":2376,"prompt_tokens":1047,"completion_tokens":1329,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":1259}},"tokens_in":663,"tokens_out":1329,"duration_ms":14994,"temperature":1.0,"reasoning_tokens":1259,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:42:03.639165+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct, for some $r\\ge 3$, $s\\ge 2$, $1\\le t\\le r-1$, and $p>1$, a $P^r_{2s}$-free $r$-graph on $n$ vertices with $\\|H\\|_{t,p} > \\|E_{n,r,s}\\|_{t,p}$ for arbitrarily large $n$; a concrete starting point is a full $(s-1)$-star with outside edges whose pairwise intersections contain two disjoint pairs, which would violate the $2$-intersecting cleaning conclusion. For the matching-number claims, a counterexample would be a family with matching number at most $s$ that is not contained in any $s$-star yet has norm exceeding $\\|H_{n,r,s}\\|_{t,p}$.","supporting_citations":[{"cited_title":"Ahlswede and L","cited_arxiv_id":null,"evidence_quote":"Supplies the complete nontrivial-intersection theorem used to bound the norm of non-star $k$-intersecting families in Lemma 4.1."},{"cited_title":"Convex Transference for Degree Powers in Extremal Set Systems","cited_arxiv_id":"2607.28616","evidence_quote":"Gives the recent degree-power results for intersecting families that the $k$-star theorems extend and sharpen."},{"cited_title":"Kostochka, D","cited_arxiv_id":null,"evidence_quote":"Provides the path Turán asymptotics, edge stability, and the two cleaning lemmas (5.6 and 5.7) on which Theorems 1.5 and 1.6 rest."}],"review_version":1}