REVIEW 1 major objections 6 minor 25 references
The $(t,p)$-Norm in Classical Extremal Problems
T0 review · 1 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read Star-type families uniquely maximize the $(t,p)$-norm under matching, intersection, and long-path restrictions.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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 $\|H_{n,r,s}\|_{t,p}$.
Extended reading notes
Core claim
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 $1\le t\le r-1$.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [Section 5.3, Lemmas 5.6–5.7] 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.
minor comments (6)
- [Section 1.1] 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 6] In the concluding remarks, 'For seek of simplicity' should read 'For the sake of simplicity'.
- [Abstract and throughout] There are spacing errors of the form 'Whent=r−1' and 'anr-graph'; these should be 'When t=r−1' and 'an r-graph'.
- [Theorem 1.1] 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.
- [Lemmas 5.6–5.7] 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 5.3.2] 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.
Circularity Check
No significant circularity: the derivations are anchored in external theorems, and the unproved KMV cleaning lemmas are a correctness risk, not a circular step.
full rationale
The paper's derivation chain contains no step in which a claimed prediction is equivalent to its inputs by construction. The matching-number results (Theorems 1.1 and 1.2) are proved by shifting plus induction, with the external Frankl theorem and Lemma 2.4 as independent anchors; no constant is fitted and no target inequality is assumed. The EKR-type results (Theorems 1.3 and 1.4) reduce non-trivial families to the external EKR/Hilton-Milner bounds and use coordinatewise degree comparison for star-containing families, again with no self-referential identification of the extremal objective with the extremal family beyond the actual extremal theorem. The path results (Theorems 1.5 and 1.6) derive the norm upper bound and stability from the external Kostochka-Mubayi-Verstraete theorem (Theorem 2.3) plus local weighted estimates, and then invoke the external cleaning Lemmas 5.6 and 5.7; although those lemmas are not proved in this paper and are load-bearing for the even-path characterization, reliance on an externa result is not circularity. Theorem 1.3 is used inside the proof of Theorem 1.6 only after it has been proved independently in Section 4, so this is legitimate reuse, not question-begging. The declarations in Section 6 and the AI-use note do not assert any assumption of the target results. Accordingly, the appropriate finding is no significant circularity, with a score of 0.
Assumptions & free parameters
assumptions (5)
- standard math Frankl's Erdős matching theorem (Theorem 2.1).
- standard math Erdős-Ko-Rado theorem and the non-trivial EKR bound (Theorem 2.2).
- standard math Kostochka-Mubayi-Verstraete path theorem (Theorem 2.3).
- domain assumption Cleaning lemmas 5.6 and 5.7 from [20, Section 6.4].
- standard math Standard shifting properties for r-graphs.
Cite this review
Pith. "Pith review of The $(t,p)$-Norm in Classical Extremal Problems." pith.science (2026). https://pith.science/paper/FJ7PZV5U
@misc{pith2026260804615,
author = {Pith},
title = {Pith review of: The $(t,p)$-Norm in Classical Extremal Problems},
year = {2026},
howpublished = {\url{https://pith.science/paper/FJ7PZV5U}},
note = {Machine review of arXiv:2608.04615}
}
abstract
Given integers $r>t\ge1$ and a real number $p>0$, the $(t,p)$-norm $||\mathcal{H}||_{t,p}$ of an $r$-graph $\mathcal{H}$ is the sum of the $p$-th powers of the degrees $d_{\mathcal{H}}(T)$ over all $t$-subsets $T\subseteq V(\mathcal{H})$. When $t=r-1$, this is the codegree $p$-norm. For all sufficiently large $n$, we obtain the following results. The first two apply in both the convex range $p>1$ and the concave range $0<p<1$. First, for $r$-graphs with matching number at most $s$, we determine the maximum $(t,p)$-norm. Second, for $k$-intersecting families, we establish an Erd\H{o}s--Ko--Rado-type theorem for the $(t,p)$-norm. Third, for $P_\ell^r$-free hypergraphs, we determine the maximum $(t,p)$-norm for every $1\le t\le r-1$ and $p>1$. In each of the three settings, we also characterize all extremal families.
Reference graph
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