{"id":"bf34f466-4ac0-48fd-ad0d-57d8addf448b","arxiv_id":"2411.15579","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For r-uniform hypergraphs, the p-norm Turán number ex_p(n,F) switches from a pseudorandom (n^{1+pα}) to a star-like (n^{p(r-1)}) regime at p = 1/(r-1-α), and the threshold case is solved for short even cycles and degree-bounded bipartite graphs.","lead":"This paper proves a phase transition for the maximum sum of degree powers in graphs and hypergraphs that avoid a fixed subgraph. It also removes a logarithmic factor from known bounds for several concrete forbidden graphs, and offers a new proof technique based on degree regularization.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's large-p regime is false for r >= 3: the star lower bound is ~ n^{p(r-1)}, while the theorem claims O((n/(r-1))^p); the proof of Claim 4.4 replaces binom(n, r-1) by n/(r-1).","rationale":"The reader's stated weakest assumption was the external growth-rate hypothesis ex(n,F)=O(n^{1+alpha}). That is a legitimate limitation, but it is not the main vulnerability: Theorem 1.2 is conditional by design, and conditional theorems are acceptable. The genuinely load-bearing problem is internal and falsifiable: for r >= 3 the stated upper bound in the star-like regime has the wrong exponent and is contradicted by an explicit F-free construction covered by the theorem's hypotheses. The proof locates the error precisely: Claim 4.4 takes the additive term binom(n,r-1) from Proposition 2.9 and replaces it by (n/(r-1)), an equality only when r=2. This single algebraic slip destroys the proof of Proposition 4.2 for hypergraphs and would, if taken literally, also make Fact 1.1's star lower bound inconsistent with the claimed upper bound. The concrete counterexample with two disjoint triples leaves no ambiguity: with alpha=1, p=2, the star S_3(n,1) is F-free and has p-norm ~ n^4/4, while the theorem promises O(n^2). The introduction's own prediction of n^{p(r-1)} in the large-p regime shows the intended statement is probably a corrected binomial term, e.g. binom(n,r-1)^p, not (n/(r-1))^p. But the manuscript as submitted states and proves the false version, and the main claim of a hypergraph phase transition is therefore not established. The verdict should be REJECT for the current version; a corrected theorem and proof could merit reconsideration.","tokens_in":24596,"tokens_out":17661,"duration_ms":154948,"concrete_test":"Compute both sides for r=3, F={123,456}, p=2, large n. The star S_3(n,1) is F-free and has ||S_3(n,1)||_2 = C(n-1,2)^2 + (n-1)(n-2)^2 ~ n^4/4. Evaluating this at n=100 gives about 6.0e7, while the Theorem 1.2 upper bound (tau_part(F)-1+o(1))(n/2)^2 is about 2500. If the numerical comparison confirms this gap for n >= 6, the theorem statement is false as written. An additional check: re-derive Claim 4.4 without replacing binom(n,r-1) by n/(r-1); the resulting additive term is (s1-1) C(n,r-1), not (s1-1)n/(r-1).","verdict_should_be":"REJECT","load_bearing_attack":"As stated, Theorem 1.2 cannot hold for r >= 3. Take F = {abc, def}, two vertex-disjoint 3-sets. This family is 3-partite with tau_part(F) = 2 and ex(n,F) = C(n-1,2) = Theta(n^2), so the hypothesis holds with alpha = 1 in [r-2, r-1) = [1,2). For p = 2 > 1/(r-1-alpha) = 1, the star S_3(n,1) = {triples containing vertex 1} is F-free, since all its edges share vertex 1 and F requires two disjoint edges. Its p-norm is C(n-1,2)^2 + (n-1)(n-2)^2 ~ n^4/4, whereas Theorem 1.2 predicts ex_p(n,F) <= (tau_part(F)-1+o(1))(n/(r-1))^p = (1+o(1))(n/2)^2 ~ n^2/4. This is a contradiction for all large n. The same error appears in the proof: in Claim 4.4, Proposition 2.9 contributes the additive term (s1-1) * binom(n, r-1), but the next line bounds this by (s1-1) * (n/(r-1)). For r=3 this replaces C(n,2) ~ n^2/2 by n/2, losing a factor n. Since all degrees in U can be as large as n^{r-1}, the subsequent replacement of d_H(v)^p by d_H(v)(n/(r-1))^{p-1} in the p-norm estimate is also invalid for r >= 3. The introduction's own heuristic, predicting n^{p(r-1)} in the large-p regime, agrees with the star lower bound and indicates that the stated (n/(r-1))^p is a typo, but as written the central theorem is false.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the p-norm Turán number ex_p(n,F), the maximum of the sum of p-th powers of vertex degrees over F-free r-graphs on n vertices. It claims a phase transition for degenerate families F satisfying ex(n,F)=O(n^{1+alpha}): for 1<p<1/(r-1-alpha) it gives ex_p(n,F)=O(n^{1+p alpha}), while for p>1/(r-1-alpha) it claims ex_p(n,F) is at most (tau_part(F)-1+o(1))(n/(r-1))^p. At the threshold it proves an O(n^{p(r-1)} log n) bound, and it confirms the conjectured removal of the log factor for several bipartite graph families, including short even cycles and one-side degree-bounded graphs. The proofs use a p-norm regularization lemma, semibipartite reductions, and dependent random choice. The graph case r=2 is also re-proved with an improved constant.","tokens_in":24987,"tokens_out":20010,"duration_ms":177774,"significance":"If the main theorem were correct, it would be a substantial extension of the Füredi-Kündgen graph result to hypergraphs and would settle the log-factor conjecture at the threshold for several natural graph families. The paper contains potentially useful tools, especially the p-norm regularization lemma and the explicit bounds in Theorem 1.4. However, the central hypergraph statement is false: Theorem 1.2 as stated is contradicted by an elementary construction for r>=3. The advertised hypergraph phase transition therefore cannot stand, and the remaining graph-case results, though possibly valid, are a much smaller contribution than the paper claims.","major_comments":[{"comment":"Theorem 1.2 is false for r>=3. Let F={abc, def} be the 3-graph consisting of two disjoint triples. F is 3-partite, hence degenerate, and ex(n,F)=Theta(n^2) for n large, since the largest intersecting 3-graph has about C(n-1,2) edges. Thus the hypothesis holds with r=3 and alpha=1, and tau_part(F)=2. For p=2>1/(r-1-alpha)=1, the theorem predicts ex_2(n,F) is at most (1+o(1))(n/2)^2 ~ n^2/4. But the star S_3(n,1), consisting of all triples containing vertex 1, is F-free and has p-norm C(n-1,2)^2+(n-1)(n-2)^2 ~ n^4/4, so ex_2(n,F)=Omega(n^4). Even the paper's own Corollary 2.5, Eq. (1), gives ex_2(n,F) >= n(3 ex(n,F)/n)^2 = Omega(n^3), already contradicting the claimed O(n^2) upper bound. The introduction's heuristic predicting n^{p(r-1)} in the large-p regime points to the same conclusion, so the error is not merely a typo in one constant.","section":"Theorem 1.2 / Proposition 4.2"},{"comment":"The proof of the large-p regime invokes Proposition 2.9 in the wrong direction. The text says that because F is contained in the complete r-partite r-graph K^r_{s1,...,sr} and S is F-free, Proposition 2.9 bounds |S|. This implication is backwards: if F is a subgraph of K, then every K-free graph is F-free, so an upper bound for K-free graphs does not upper-bound F-free graphs. For the family F={abc, def} from the previous comment, F is a proper subgraph of K^3_{2,2,2}, and F-free semibipartite graphs can have far more edges than K^3_{2,2,2}-free graphs; for example, a star with one high-degree vertex in U is F-free. Consequently the O(n) bound on the sum of degrees of high-degree vertices in Eq. (10) is unjustified.","section":"Claim 4.4"},{"comment":"Even granting Proposition 2.9, the absorption of error terms in Claim 4.4 is not justified in the stated parameter range. The assertion |S2| <= |U|^2 C(n,r-2) <= (epsilon/(6r))(n/(r-1))^p requires 2 delta_2 + r - 2 < p. The paper only guarantees delta_2 <= (p-1)/p, which does not imply this inequality; for instance, with r=4, alpha=2, and p=1.5, one has p>1/(r-1-alpha)=1, but any positive delta_2 gives 2 delta_2 + 2 > 1.5. The same obstruction occurs in the absorption of the first term of Proposition 2.9 into (epsilon/2)n/(r-1), since its exponent r-1-1/(s1...s_{r-1})+delta_2 is not always less than 1. Thus the proof of Proposition 4.2 does not establish the claimed upper bound even if the direction of the Proposition 2.9 application were fixed.","section":"Claim 4.4"}],"minor_comments":[{"comment":"In Claim 6.8 the inequality W_{\\ell+1}(H) >= 4^{\\ell-1} n^2 is obtained by replacing v(H) by n in the denominator, but the proof only gives v(H) <= 2n. This replacement is not valid and overestimates the constant by a factor 2^{\\ell-2}. The argument still works if one replaces n^{\\ell-2} by (2n)^{\\ell-2}, yielding the weaker but sufficient bound W_{\\ell+1}(H) >= 2^{\\ell} n^2, so this is a harmless constant slip.","section":"Claim 6.8"},{"comment":"The second lower bound in Fact 1.1 is not the true p-norm of the star-like r-graph S_r(n,t) when r>=3. For r=3, each center vertex has degree about n^2/2, so the p-norm is about t (n^2/2)^p, whereas the displayed expression (n/(r-1))^p is only t (n/2)^p. The displayed inequality is true but far weaker than the actual star construction, and the surrounding text should not attribute it to the p-norm of S_r(n,t).","section":"Fact 1.1"}],"recommendation":"reject","confidential_remarks":"I agree with the sceptic's counterexample: the central theorem for r>=3 is false, and the proof of Proposition 4.2 contains a logical error in the use of Proposition 2.9. A revision restricted to the graph case r=2 would avoid the counterexample but would be a much smaller paper, largely overlapping with known results of Füredi and Kündgen. I cannot recommend acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: the central hypergraph theorem, Theorem 1.2, is false as stated for r≥3. Take F = {abc, def}, two disjoint triples. This family is 3-partite with τ_part(F)=2 and ex(n,F)=Θ(n^2), so α=1. For p=2>1/(r-1-α)=1, the theorem predicts ex_2(n,F) ≤ (1+o(1))(n/2)^2. But the star S_3(n,1) (all triples containing vertex 1) is F-free, and its p-norm is ~ n^4/4, a contradiction. The introduction's own heuristic, which says the large-p regime should be O(n^{p(r-1)}), agrees with the star lower bound. So the theorem statement is not a harmless typo; the proof of Proposition 4.2 collapses too. Claim 4.4 bounds the edges with exactly one high-degree vertex by (s1−1)(n/(r−1)), but the first term from Proposition 2.9, of order |U| n^{r−1−1/(s1⋯s_{r-1})}, is not negligible for r≥3. The same linear term appears in Proposition 2.9, which looks misstated: for m=1, r=3, s1=s2=s3=2, it gives O(n^{7/4}), while the star with one left vertex has ~n^2/2 edges and is K_{2,2,2}-free. Fact 1.1's second lower bound is also wrong for r≥3, since the star degree is n^{r-1}, not n. This is a systematic dimensional error from carrying the r=2 formula into hypergraphs.\n\nWhat is worth keeping: Lemma 3.1, the p-norm regularization lemma, is new and looks like a genuinely useful tool. Theorem 1.3, the log n bound at the threshold, seems plausible and its proof does not rely on the broken Claim 4.4. The r=2 results are essentially Füredi–Kündgen, which the authors honestly credit, and the alternative proof that yields constants is a reasonable contribution. Theorem 1.4 (graph only) appears sound; the reader's minor concerns about Claim 6.8 and the bipartite reduction are minor.\n\nThis is not a paper to send to review in its current form. The advertised hypergraph phase transition is false, and fixing it requires reworking the large-p argument and the statement itself. If the authors restrict to r=2 or repair the hypergraph case, there is a salvageable core, but as it stands the main result is a counterexample away from being true. I would not put this on a reading group either.","headline":"Theorem 1.2 is false for r≥3: the large-p bound should be n^{p(r-1)}, not (n/(r-1))^p, and the proof repeats the same dimensional error in Claim 4.4.","tokens_in":25613,"tokens_out":15913,"would_cite":false,"duration_ms":128752,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C65","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The p-norm Turán number of a degenerate hypergraph family switches sharply at p = 1/(r-1-α), with a tight star constant above and pseudorandom growth below.","keywords":["degenerate Turán problem","degree powers","p-norm","phase transition","hypergraphs","partition number","even cycles","regularization"],"falsifier":"Compute the p-norm Turán number for the 3-graph F from Section 7 with $\\tau_{\\mathrm{ind}}(F)=3$ and $\\tau_{\\mathrm{part}}(F)=4$, for $p>2$. If F-free 3-graphs on $n$ vertices can have p-norm exceeding $(3+\\varepsilon)(n/2)^p$ infinitely often, the supercritical bound of Theorem 1.2 is false; the open question is whether the star lower bound $(\\tau_{\\mathrm{ind}}(F)-1+o(1))(n/2)^p$ can be pushed closer to the $\\tau_{\\mathrm{part}}(F)-1$ upper bound.","tokens_in":24331,"feed_emoji":"🔀","tokens_out":11505,"duration_ms":92748,"temperature":0.7,"pith_summary":"This paper proves a sharp phase transition for p-norm Turán numbers of degenerate families of r-uniform hypergraphs. For a forbidden family F whose ordinary Turán number grows at most like $n^{1+\\alpha}$ with $\\alpha < r-1$, the maximum p-norm of an F-free r-graph is $O(n^{1+p\\alpha})$ when $p$ is below the threshold $1/(r-1-\\alpha)$, and at most $(\\tau_{\\mathrm{part}}(F)-1+o(1))(n/(r-1))^p$ above it. In the graph case $r=2$, the supercritical bound is asymptotically tight because the partition number $\\tau_{\\mathrm{part}}(F)$ equals the independent covering number $\\tau_{\\mathrm{ind}}(F)$. The paper also removes the logarithmic factor conjectured removable by Füredi and Kündgen at the threshold for several families, including short even cycles and s-bounded bipartite graphs. These results matter because the p-norm interpolates between edge count and maximum degree, so the threshold describes how extremal constructions change as the objective shifts from counting edges to concentrating them.","feed_headline":"Sharp threshold proved for degree-power Turán numbers","feed_subtitle":"Below the threshold pseudorandom graphs rule; above it star-like hypergraphs give the exact leading constant.","key_machinery":"The load-bearing tool is a p-norm version of the classical $\\Delta$-almost-Regularization Theorem of Erdős–Simonovits, stated as Lemma 3.1. Given an r-graph with p-norm at least $C n^{1+p\\alpha}$, this lemma produces a subgraph $H$ on $m$ vertices that keeps a $(1-\\varepsilon)$ fraction of the p-norm, has maximum degree at most a constant times $(\\|H\\|_p/m)^{1/p}$, and has more than $\\hat{C} m^{1+\\alpha}$ edges. This converts an F-free graph with overly large p-norm into an F-free graph with too many edges, contradicting the assumed bound $\\mathrm{ex}(n,F)=O(n^{1+\\alpha})$. In the supercritical regime, the proof instead uses a semibipartite upper bound for complete r-partite hypergraphs (Proposition 2.9 from [HHL+23]) together with the partition number $\\tau_{\\mathrm{part}}(F)$, and it bootstraps the subcritical result at a smaller exponent $\\hat{p}<p$. The star-like hypergraph $S_r(n,t) = \\{ e \\in \\binom{[n]}{r} : |e \\cap [t]| = 1 \\}$ supplies the matching lower bound with $t = \\tau_{\\mathrm{part}}(F)-1$.","core_discovery":"On the paper's own terms, the central result is Theorem 1.2: if F is a degenerate family of r-graphs with $\\mathrm{ex}(n,F)=O(n^{1+\\alpha})$ for some constant $\\alpha \\in [r-2, r-1)$, then for every $p>1$ there is a constant $C_F$ with $\\mathrm{ex}_p(n,F) \\le C_F n^{1+p\\alpha}$ for $1<p<1/(r-1-\\alpha)$, and $\\mathrm{ex}_p(n,F) \\le (\\tau_{\\mathrm{part}}(F)-1+o(1))(n/(r-1))^p$ for $p>1/(r-1-\\alpha)$. Here $\\tau_{\\mathrm{part}}(F)$ is the minimum, over r-partite members of F, of the smallest class size in a partition of the vertex set into r classes each met by every edge exactly once. The two regimes correspond to pseudorandom, almost regular constructions below the threshold and to star-like hypergraphs $S_r(n, \\tau_{\\mathrm{part}}(F)-1)$ above it. For $r=2$, the parameters $\\tau_{\\mathrm{ind}}$ and $\\tau_{\\mathrm{part}}$ coincide, so the paper obtains the exact leading coefficient in the star regime, going beyond the earlier proof of Füredi and Kündgen. The paper also establishes Theorem 1.3, a general $O(n^{p^*(r-1)}\\log n)$ bound at the threshold $p^*=1/(r-1-\\alpha)$, and Theorem 1.4, which removes the logarithmic factor for families of short even cycles and for s-bounded bipartite graphs.","pith_inferences":["If the Erdős–Simonovits Rational Exponent Conjecture holds for a graph family, the subcritical exponent $n^{1+p\\alpha}$ is tight, so the phase transition would give a complete piecewise-linear description of the p-norm exponent in terms of the ordinary Turán exponent.","For $r \\ge 3$, the gap between $\\tau_{\\mathrm{ind}}(F)$ and $\\tau_{\\mathrm{part}}(F)$ leaves open whether the supercritical limit exists; the paper's own Problem 7.1 asks exactly this, and the 3-graph example with $\\tau_{\\mathrm{ind}}=3$, $\\tau_{\\mathrm{part}}=4$ is the natural first test case.","The p-norm regularization lemma is a standalone tool that could be applied to other degree-based objectives, such as counting copies of a fixed subgraph or studying the $(t,p)$-norm Turán numbers mentioned in the concluding remarks, by replacing the edge-count condition with the relevant count.","A concrete testable prediction of the phase transition: for complete bipartite graphs $K_{s,t}$ with $t$ large, where $\\mathrm{ex}(n,K_{s,t})=\\Theta(n^{2-1/s})$, the p-norm extremal number should be $\\Theta(n^{1+p(1-1/s)})$ for every $1<p<s$."],"forward_implications":["For any degenerate family F satisfying $\\mathrm{ex}(n,F)=O(n^{1+\\alpha})$, the p-norm Turán number is determined up to a constant factor in the subcritical regime and asymptotically in the supercritical regime, for every $p>1$.","In the graph case, the exact leading constant $\\mathrm{ex}_p(n,F)=(\\tau_{\\mathrm{ind}}(F)-1+o(1))n^p$ for $p>1/(1-\\alpha)$ follows from the theorem together with the star construction.","The threshold bound at $p=p^*$ extends Füredi–Kündgen's log-factor conjecture to all uniformity ranks $r$, and the families in Theorem 1.4 now satisfy the conjectured bound without the logarithmic factor.","For $\\{C_4,\\dots,C_{2\\ell}\\}$, the bound $\\mathrm{ex}_{\\ell/(\\ell-1)}(n,\\{C_4,\\dots,C_{2\\ell}\\}) \\le 765 n^{\\ell/(\\ell-1)}$ holds, and for s-bounded bipartite F, $\\mathrm{ex}_s(n,F) \\le 2(|V(F)|^s/s! + |V(F)|) n^s$ holds.","If $\\mathrm{ex}(n,F)=O(n^{1+\\beta})$ with $\\beta \\le r-2$, the theorem implies the star-like bound $(\\tau_{\\mathrm{part}}(F)-1+o(1))(n/(r-1))^p$ for every $p \\ge 1$, so no subcritical regime exists."],"supporting_citations":[{"why":"introduced the p-norm threshold for bipartite graphs and conjectured the log-factor removal; the present paper extends and strengthens this result.","marker":"[FK06]"},{"why":"survey containing the classical $\\Delta$-almost-Regularization Theorem (their Theorem 2.19) whose p-norm adaptation is Lemma 3.1.","marker":"[FS13]"},{"why":"supplies Proposition 2.9, the semibipartite upper bound for complete r-partite hypergraphs used to control stars in the supercritical regime.","marker":"[HHL+23]"},{"why":"establishes that every degenerate family contains an r-partite member and satisfies $\\mathrm{ex}(n,F)=O(n^{r-\\delta})$, used in the supercritical proof.","marker":"[Erd64b]"},{"why":"the classical Kővári–Sós–Turán bound that Proposition 2.9 generalizes to r-partite r-graphs.","marker":"[KST54]"},{"why":"gives the $O(n^{1+1/\\ell})$ bound for graphs without $C_4,\\dots,C_{2\\ell}$, used in the proof of Theorem 1.4(i).","marker":"[LV05]"},{"why":"gives the bipartite Zarankiewicz-type bound for short even cycles used in the proof of Theorem 1.4(i).","marker":"[NV05]"}],"fun_headline_variants":["Exact constant for hypergraph Turán p-norms in star regime","Beyond Füredi–Kündgen: sharp threshold with exact constant","Pseudorandom vs star-like: sharp threshold for p-norm Turán","Log factor dropped for short even cycles in p-norm Turán"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem assumes that the ordinary Turán number of the forbidden family is polynomially bounded as $O(n^{1+\\alpha})$ with $\\alpha < r-1$; for many natural families, such as long even cycles, this polynomial exponent is still open, so the phase-transition conclusion applies only to families for which that external bound has been proved.","fun_headline_variants_meta":{"raw":{"variants":["Exact constant for hypergraph Turán p-norms in star regime","Beyond Füredi–Kündgen: sharp threshold with exact constant","Pseudorandom vs star-like: sharp threshold for p-norm Turán","Log factor dropped for short even cycles in p-norm Turán"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000293,"raw_usage":{"total_tokens":1828,"prompt_tokens":1186,"completion_tokens":642,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":802,"completion_tokens_details":{"reasoning_tokens":564}},"tokens_in":802,"tokens_out":642,"duration_ms":6016,"temperature":1.0,"reasoning_tokens":564,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:10:02.330549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the p-norm Turán number for the 3-graph F from Section 7 with $\\tau_{\\mathrm{ind}}(F)=3$ and $\\tau_{\\mathrm{part}}(F)=4$, for $p>2$. If F-free 3-graphs on $n$ vertices can have p-norm exceeding $(3+\\varepsilon)(n/2)^p$ infinitely often, the supercritical bound of Theorem 1.2 is false; the open question is whether the star lower bound $(\\tau_{\\mathrm{ind}}(F)-1+o(1))(n/2)^p$ can be pushed closer to the $\\tau_{\\mathrm{part}}(F)-1$ upper bound.","supporting_citations":[],"review_version":1}