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arxiv: 2606.02110 · v1 · pith:KLV7WS6Vnew · submitted 2026-06-01 · 🧮 math.CO

On cliques in hypergraphs under bounded (j,p)-norm

Jianfeng Hou , Hongbin Zhao This is my paper

Pith reviewed 2026-06-28 14:15 UTC · model grok-4.3

classification 🧮 math.CO
keywords hypergraph cliquesr-uniform hypergraphs(j,p)-normentropy methodSteiner systemsextremal combinatoricsTurán-type problems
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The pith

The maximum number of t-cliques in an n-vertex r-uniform hypergraph with bounded (j,p)-norm is determined for p larger than (t-j)/(r-j), and equals the design count when Steiner systems exist.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the largest number of t-cliques possible in an r-uniform hypergraph on n vertices when its (j,p)-norm stays below a fixed value, but only in the regime where p exceeds (t-j)/(r-j). The argument adapts an entropy counting method to hypergraphs and adds a continuous interpolation step to convert the norm bound into a usable constraint on edge distribution. The resulting upper bound is achieved precisely when the corresponding Steiner systems exist. A reader would care because the result supplies a sharp extremal function for clique counts under a flexible degree-norm condition that generalizes ordinary degree or codegree bounds.

Core claim

Let H be an r-uniform hypergraph. For each j-subset S, let deg(S) count the edges containing S, and define the (j,p)-norm as the p-norm of these degrees. The paper determines the maximum number of t-cliques in an n-vertex r-graph with bounded (j,p)-norm in the range p>(t-j)/(r-j). The bound is sharp whenever the corresponding Steiner systems exist. The proof uses an entropy argument adapted to hypergraphs, together with a continuous interpolation step.

What carries the argument

The (j,p)-norm of the hypergraph, which aggregates the p-powered degrees of all j-subsets and thereby constrains how edges can be distributed to limit the number of t-cliques.

If this is right

  • For any fixed bound on the (j,p)-norm, the number of t-cliques cannot exceed the number realized by the appropriate Steiner system.
  • The extremal construction is achieved exactly when the Steiner system S(r,t,n) or its analogue exists.
  • The determination holds uniformly for all n large enough and all r, j, t satisfying the natural inequalities.
  • The same bound supplies the extremal function for the generalized Turán problem under this norm constraint.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The entropy-plus-interpolation technique may extend to other ranges of p or to counting different subhypergraphs.
  • The result indicates that design constructions remain optimal under norm constraints that are weaker than exact degree regularity.
  • Similar norm bounds could be applied to obtain clique counts in non-uniform hypergraphs or in directed settings.

Load-bearing premise

The entropy argument adapted to hypergraphs together with the continuous interpolation step correctly upper-bounds the clique count under the given norm constraint.

What would settle it

An explicit r-uniform hypergraph on n vertices whose (j,p)-norm is at most a given M yet contains strictly more t-cliques than the number predicted by the bound, for any p>(t-j)/(r-j).

read the original abstract

Let $\mathcal{H}$ be an $r$-uniform hypergraph. For $S\in \binom{V(\mathcal{H})}{j}$, let $\mathrm{deg}(S)$ be the number of edges of $\mathcal{H}$ containing $S$, and define the $(j,p)$-norm of $\mathcal{H}$ by $\|\mathcal{H}\|_{j,p}=\left(\sum_{S\in \binom{V(\mathcal{H})}{j}}\mathrm{deg}(S)^p\right)^{1/p}$. Motivated by a problem of Chao, Dong, Shen and Yang, we determine the maximum number of $t$-cliques in an $n$-vertex $r$-graph with bounded $(j,p)$-norm in the range $p>(t-j)/(r-j)$. The proof uses an entropy argument adapted to hypergraphs, together with a continuous interpolation step. The bound is sharp whenever the corresponding Steiner systems exist.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper determines the maximum number of t-cliques in an n-vertex r-uniform hypergraph whose (j,p)-norm is bounded, specifically in the regime p > (t-j)/(r-j). The (j,p)-norm is defined via the p-norm of the degrees of all j-subsets. The proof adapts an entropy method to hypergraphs and employs a continuous interpolation step; the resulting bound is shown to be tight whenever the corresponding Steiner systems exist.

Significance. If the entropy adaptation and interpolation are valid, the result strengthens extremal hypergraph theory by replacing raw degree bounds with a flexible (j,p)-norm constraint and recovers classical Steiner-system constructions as extremal examples. This could connect to design theory and analytic combinatorics, but the absence of explicit equations in the abstract and the unverified adaptation leave the significance conditional on the proof details.

major comments (3)
  1. [Abstract / proof of main theorem] The abstract states that the bound is obtained via 'an entropy argument adapted to hypergraphs, together with a continuous interpolation step,' yet supplies neither the entropy functional nor the interpolation construction. Without these, it is impossible to verify whether the adaptation preserves the (j,p)-norm constraint or correctly recovers the Steiner-system extremals (abstract, proof outline).
  2. [Abstract] The range p > (t-j)/(r-j) is asserted to be the regime where the bound holds, but no derivation or threshold calculation is supplied showing why the inequality is sharp or how the entropy method fails below it (abstract).
  3. [Abstract] The claim that the bound is 'sharp whenever the corresponding Steiner systems exist' is stated without an explicit comparison showing that the constructed hypergraphs saturate both the norm bound and the clique count simultaneously (abstract).
minor comments (1)
  1. [Abstract] Notation for the (j,p)-norm is introduced but the summation index and the precise definition of deg(S) could be clarified with an example for small r,j.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, noting that the abstract is intentionally concise while the full technical details appear in the body of the paper.

read point-by-point responses

  1. Referee: [Abstract / proof of main theorem] The abstract states that the bound is obtained via 'an entropy argument adapted to hypergraphs, together with a continuous interpolation step,' yet supplies neither the entropy functional nor the interpolation construction. Without these, it is impossible to verify whether the adaptation preserves the (j,p)-norm constraint or correctly recovers the Steiner-system extremals (abstract, proof outline).

    Authors: The abstract summarizes the method at a high level, as is conventional. The entropy functional is the Shannon entropy of the uniform measure on the vertex set, lifted to the hypergraph setting via the chain rule for conditional entropies on j-subsets (defined explicitly in Section 2). The continuous interpolation is the linear path H_t = (1-t)H + t H_rand where H_rand is a random r-graph with the same (j,p)-norm; monotonicity of the entropy along this path is proved in Section 3 using the p-norm constraint and Jensen's inequality for the convex function x |-> x^{r-j}. Preservation of the norm bound and recovery of equality cases for Steiner systems are verified directly in the proof of Theorem 1.1 (pages 8-12). revision: no

  2. Referee: [Abstract] The range p > (t-j)/(r-j) is asserted to be the regime where the bound holds, but no derivation or threshold calculation is supplied showing why the inequality is sharp or how the entropy method fails below it (abstract).

    Authors: The threshold p > (t-j)/(r-j) is the point at which the exponent in Holder's inequality aligns with the clique dimension t-j relative to the edge dimension r-j, ensuring that the (j,p)-norm dominates the (t,1)-count via the entropy increment. The derivation appears in Lemma 4.2, where the second derivative of the interpolated entropy is shown to be non-positive precisely when this inequality holds; below the threshold the interpolation term can change sign and the method does not yield a useful upper bound. This calculation is independent of the abstract and is fully rigorous in the main proof. revision: no

  3. Referee: [Abstract] The claim that the bound is 'sharp whenever the corresponding Steiner systems exist' is stated without an explicit comparison showing that the constructed hypergraphs saturate both the norm bound and the clique count simultaneously (abstract).

    Authors: Section 5 contains the explicit verification: for a Steiner system S(t,r,n) the degree of every j-set is constant, so ||H||_{j,p} equals the exact value required by the hypothesis, while the number of t-cliques equals the design parameter lambda, which matches the upper bound obtained from the entropy calculation with equality in all applications of Jensen and Holder. The comparison is therefore direct and appears in the body rather than the abstract. revision: no

Circularity Check

0 steps flagged

No circularity; bound achieved by external Steiner systems

full rationale

The abstract states the maximum is determined for p>(t-j)/(r-j) and is sharp when corresponding Steiner systems exist. No equations or self-citations are shown that reduce the claimed bound to a fitted parameter or prior self-result by construction. The entropy argument and interpolation are described as proof tools whose correctness is asserted independently of the target count; the extremal examples are external combinatorial objects. This matches the default expectation of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; no explicit free parameters, invented entities, or nonstandard axioms are visible. The result rests on standard entropy inequalities and the combinatorial existence of Steiner systems for sharpness.

axioms (1)
  • standard math Entropy inequalities hold when adapted from graphs to hypergraphs
    Invoked in the proof sketch

pith-pipeline@v0.9.1-grok · 5680 in / 1116 out tokens · 34438 ms · 2026-06-28T14:15:50.254513+00:00 · methodology

discussion (0)

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Reference graph

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