Let \(\mathcal{H}\) be an \(r\)-uniform hypergraph. For \(S\in \binom{V(\mathcal{H})}{j}\), let \(\deg(S)\) be the number of edges of \(\mathcal{H}\) containing \(S\), and define the \((j,p)\)-norm of \(\mathcal{H}\) by \[\|\mathcal{H}\|_{j,p}=\left(\sum_{S\in \binom{V(\mathcal{H})}{j}}\deg(S)^p\right)^{1/p}.\] 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.
For a graph \(G\), let \(k_t(G)\) denote the number of \(t\)-cliques \(K_t\) in \(G\). Estimating \(k_t(G)\) under natural edge-density conditions is a classical problem in extremal graph theory. Turán’s theorem implies that any graph on \(n\) vertices with more edges than the Turán graph contains a copy of \(K_{t}\). More generally, determining the minimum possible value of \(k_t(G)\) among graphs whose edge density exceeds the Turán threshold is the clique supersaturation problem. This problem was completely resolved through the work of Razborov [1] for triangles, Nikiforov [2] for \(4\)-cliques, and Reiher [3], who proved the Lovász–Simonovits clique density conjecture [4] in full generality.
Another direction is to maximize \(k_t(G)\) over graphs \(G\) subject to specific parameter constraints. A typical example concerns graphs with a fixed number of edges, where the extremal maximum is precisely governed by the classical Kruskal–Katona theorem [5], [6]. At the other extreme, when both the number of vertices and the maximum degree \(\Delta\) of \(G\) are fixed, Gan, Loh, and Sudakov [7] conjectured that \(k_t(G)\) is maximized by a disjoint union of cliques of size \(\Delta+1\) (together with at most one smaller clique). This conjecture, which reformulates earlier questions on independent sets [8]–[10], was fully resolved by Chase [11]; subsequently, Chao and Dong [12] provided an alternative proof.
The number of edges and the maximum degree of a graph correspond naturally to the \(\ell_1\)-norm and the \(\ell_\infty\)-norm of its degree sequence, respectively. This suggests the study of extremal problems under \(\ell_p\)-norm constraints on the degree sequence. For a real number \(p > 0\), the \(p\)-norm of \(G\), denoted by \(||G||_p\) is defined as \[\begin{align} ||G||_p = \left( \sum_{v \in V} \deg(v)^p \right)^{1/p}, \end{align}\] where \(\deg(v)\) denotes the degree of \(v\) in \(G\).
The systematic study of \(\ell_p\)-norms in extremal graph theory originated in Turán-type problems. While classical Turán problems ask for the maximum number of edges in an \(n\)-vertex graph with prescribed forbidden subgraphs, equivalently maximizing the \(\ell_1\)-norm of the degree sequence, Caro and Yuster [13], [14] extended this framework to \(\ell_p\)-norms. Erdős–Stone-type results in this setting were subsequently established by Bollobás–Nikiforov [15].
Recently, Chao, Dong, Shen, and Yang [16] investigated the maximum number of \(t\)-cliques in graphs with bounded \(\ell_p\)-norm, establishing asymptotically sharp upper bounds both with and without a fixed number of vertices.
In this note, we study the maximum number of \(t\)-cliques in hypergraphs with a fixed number of vertices and bounded \((j,p)\)-norm. For \(r\ge 2\) an \(r\)-uniform hypergraph (henceforth an \(r\)-graph) \(\mathcal{H}\) is a collection of \(r\)-subsets of some finite set \(V\). For an integer \(j\) with \(1\le j <r\) and a real number \(p > 0\), the degree of a \(j\)-subset \(S \in \binom{V}{j}\), denoted by \(\deg(S)\), is defined as the number of edges in \(\mathcal{H}\) containing \(S\). Following the definitions in [16], [17], the \((j, p)\)-norm of \(\mathcal{H}\) is defined as \[\begin{align} \|\mathcal{H}\|_{j,p} = \left( \sum_{S \in \binom{V}{j}} \deg(S)^p \right)^{1/p}. \end{align}\] Note that if \(j = r - 1\), this framework seamlessly recovers the \(\ell_p\)-norm \(\|\mathcal{H}\|_p\) of \(\mathcal{H}\).
Similarly, the study of \((j,p)\)-norms in hypergraphs originated in Turán-type problems. Balogh, Clemen, and Lidický [18], [19] initiated this direction by investigating Turán-type problems for \(3\)-graphs under \(\ell_2\)-norm constraints. Notably, they resolved the Turán tetrahedron problem under the \(\ell_2\)-norm using computer–assisted flag algebra computations, a powerful tool introduced by Razborov [20]. Subsequently, Chen, Iľkovič, León, Liu, and Pikhurko [17] initiated a systematic study of Turán-type problems under general \((j,p)\)-norm constraints. See [21]–[27] for further results in this active direction.
A natural extension of Turán-type problems is to estimate the number of \(t\)-cliques \(k_t^r(\mathcal{H})\) in an \(r\)-graph \(\mathcal{H}\) with bounded \((j,p)\)-norm. Very recently, Chao, Dong, Shen, and Yang [16] established a tight upper bound for \(k_t(\mathcal{H})\) under a fixed \((j,p)\)-norm.
To state their result, we first introduce some notation. For a real number \(x\ge 0\), define \[\begin{align} \binom{x}{k} = \frac{x(x-1)\cdots(x-k+1)}{k!}. \end{align}\] When \(p>\frac{t-j}{r-j}\), let \[\begin{align} \tilde{h}(x) = \frac{\binom{x}{t}}{\binom{x}{j}\binom{x-j}{r-j}^p}. \end{align}\] Let \(\widetilde{s}_R\) be the unique real number maximizing \(\widetilde{h}(x)\) on the interval \((t-1,\infty)\). As shown by Chao, Dong, Shen, and Yang [16], such a number exists and is unique. Moreover, \(\widetilde{h}(x)\) is strictly increasing on \((t-1,\widetilde{s}_R]\) and strictly decreasing on \([\widetilde{s}_R,\infty)\).
Theorem 1 (Chao–Dong–Shen–Yang [16]). Suppose \(C > 0\). Let \(H\) be an \(r\)-uniform hypergraph with \(\|\mathcal{H}\|_{j,p} \le C\).
If \(0 < p \le \frac{t-j}{r-j}\), then \(k_t^r(\mathcal{H}) \le \max\left\{\binom{u}{t}, 0\right\}\), where \(u\) is the positive real with \(C = \sqrt[p]{\binom{u}{j}\binom{u-j}{r-j}}\).
If \(p > \frac{t-j}{r-j}\), then \(k_t^r(\mathcal{H}) \le C^p \cdot \widetilde{h}(\widetilde{s}_R)\).
They also remarked that it would be interesting to consider the case where \(\mathcal{H}\) has a fixed number of vertices. This fixed-vertex variant is natural from the extremal point of view.
Problem 1 (Chao–Dong–Shen–Yang [16]). For integers \(r\ge 2, t> r, n\) and a real number \(C\), what is the maximum possible number of \(t\)-cliques in an \(n\)-vertex \(r\)-graph \(\mathcal{H}\) with \(\|\mathcal{H}\|_{j,p}\le C\)?
For Problem 1, when \(0 < p \le \frac{t-j}{r-j}\), the upper bound in Theorem 1 (i) continues to hold under the additional constraint that the number of vertices is fixed at \(n\). Indeed, any meaningful value of \(C\) satisfies \(C^p \le \binom{n}{j}\binom{n-j}{r-j}^p\), which by monotonicity forces \(n \ge u\). By the same argument, the bound in Theorem 1 (ii) holds when \(p > \frac{t-j}{r-j}\) and \(C^p < \binom{n}{j} \binom{\tilde{s}_{\mathbb{R}}-j}{r-j}^p\). Thus the main unresolved part of Problem 1 is the high-norm range in which the fixed vertex constraint interacts nontrivially with the local \(j\)-degree constraint. For the remaining case, we have
Theorem 2. Let \(t, r, j, n\) be fixed integers such that \(n\ge t > r > j \ge 1\), and let \(C>0\) and \(p>0\) be real numbers satisfying \[p > \frac{t-j}{r-j} \quad \text{ and } \quad C^p \ge \binom{n}{j} \binom{\tilde{s}_{\mathbb{R}}-j}{r-j}^p\] If \(\mathcal{H}\) is an \(n\)-vertex \(r\)-graph with \(\|\mathcal{H}\|_{j,p}\le C\), then \[k_t^{(r)}(\mathcal{H}) \le \frac{\binom{n}{j}}{\binom{u}{j}} \binom{u}{t},\] where \(u \ge \tilde{s}_{\mathbb{R}}\) is the unique real number satisfying \(\binom{n}{j}^{1/p} \binom{u-j}{r-j} = C\).
The main difficulty in proving Theorem 2 is that the graph argument of Chao–Dong–Shen–Yang does not extend directly to \(r\)-graphs. In the hypergraph setting, the first exposed object is naturally a \(j\)-set rather than a single vertex, and its entropy must be controlled simultaneously by the global bound \(|V(\mathcal{H})|=n\) and by the \((j,p)\)-norm of the local degrees. We overcome this by decomposing the entropy of a random ordered \(t\)-clique into a \(j\)-prefix, an \((r-j)\)-extension, and a tail, and then using a continuous interpolation argument to connect the resulting product inequalities with the monotonicity of \(\widetilde{h}\).
The following proposition shows that the upper bound in Theorem 2 is the best possible under certain number-theoretic conditions.
Proposition 2. Let \(u, t, r, j\) be fixed integers such that \(u> t > r > j \ge 1\). For sufficiently large \(n\) satisfying the necessary divisibility conditions \[\label{necessary-divisibility-condition-n} \binom{n-i}{j-i} \equiv 0 \pmod{\binom{u-i}{j-i}} \quad \text{for all } i \in \{0, 1, \dots, j-1\},\qquad{(1)}\] there exists an \(n\)-vertex \(r\)-uniform hypergraph \(\mathcal{H}^*\) with \(\|\mathcal{H}^*\|_{j,p}=\binom{n}{j}^{1/p}\binom{u-j}{r-j}\) such that \(k_t^{(r)}(\mathcal{H}^*) = \frac{\binom{n}{j}}{\binom{u}{j}} \binom{u}{t}\).
The rest of this paper is organized as follows. Section 2 introduces the necessary notation and entropy preliminaries. Section 3 prove Theorem 2 using the entropy method. Section 4 presents the extremal constructions using Steiner systems to establish Proposition 2.
We prove Theorem 2 using the entropy approach introduced by Chao–Yu [28]. This section outlines the entropy properties used throughout the paper. Let \(X\) be a discrete random variable taking values in a finite set \(\Omega\). For simplicity, let \(p_X(x) := \mathbb{P}(X=x)\) for every \(x \in \Omega\). The support of \(X\) is defined as \[\mathrm{supp(X)}\mathrel{\vcenter{:}}=\{x\in\Omega: p_X(x)>0\}.\] The Shannon entropy [29] of \(X\) is defined as \[\mathbb{H}(X)\mathrel{\vcenter{:}}= -\sum\limits_{x\in \mathrm{supp}(X)}p_X(x)\cdot \log_2{p_X(x)}.\] The following fundamental upper bound on entropy is widely used.
Proposition 3. For any discrete random variable \(X\), \(\mathbb{H}(X) \le \log_2|\mathrm{supp}(X)|\), where equality holds if and only if the distribution of \(X\) is uniform on \(\mathrm{supp}(X)\).
For any random vector \(\mathbf{X} = (X_1, \dots, X_n)\), we denote its joint entropy by \(\mathbb{H}(\mathbf{X}) := \mathbb{H}(X_1, \dots, X_n)\). For two random variables \(X\) and \(Y\), let \(X \mid Y = y\) denote the conditional random variable of \(X\) given \(Y = y\), whose entropy is denoted by \(\mathbb{H}(X \mid Y = y)\). The conditional entropy of \(X\) given \(Y\) is defined as \[\begin{align} \mathbb{H}(X \mid Y) := \sum_{y \in \mathrm{supp}(Y)} p_Y(y) \mathbb{H}(X \mid Y = y). \end{align}\] One can easily verify that \(\mathbb{H}(X \mid Y) = \mathbb{H}(X, Y) - \mathbb{H}(Y)\), which yields the following chain rule.
Proposition 4. Let \(X_1, \dots, X_n\) be discrete random variables. Then \[\mathbb{H}(X_1,\dots,X_n)=\mathbb{H}(X_1)+\mathbb{H}(X_2 \mid X_1)+\dots+\mathbb{H}(X_n \mid X_1,\dots,X_{n-1}).\]
Furthermore, we will rely on the following crucial lemma regarding the entropy of randomly ordered subsets.
Lemma 1 (Chao–Dong–Shen–Yang [16]). Let \(\mathcal{A}\) be a family of \(d\)-subsets of \([n]\). We sample a set \(A \in \mathcal{A}\) uniformly at random, and then uniformly choose a random ordering of the elements of \(A\) to form a vector \((X_1, \dots, X_d) \in [n]^d\). Then \[2^{\mathbb{H}(X_1)} \geq 2^{\mathbb{H}(X_2|X_1)} + 1 \geq 2^{\mathbb{H}(X_3|X_1, X_2)} + 2 \geq \dots \geq 2^{\mathbb{H}(X_d|X_1, \dots, X_{d-1})} + d - 1.\]
In this section, we prove Theorem 2 using ideas from [16] with a continuous interpolation step. Let \(t, r, j, n\) be fixed integers such that \(n\ge t > r > j \ge 1\), and let \(C, p>0\) be real numbers satisfying \[p > \frac{t-j}{r-j} \quad \text{ and } \quad C^p \ge \binom{n}{j} \binom{\tilde{s}_{\mathbb{R}}-j}{r-j}^p.\] Let \(\mathcal{H}\) be an \(n\)-vertex \(r\)-graph with \(\|\mathcal{H}\|_{j,p}\le C\). Suppose for the sake of contradiction that \(k_t^{(r)}(\mathcal{H}) > \frac{\binom{n}{j}}{\binom{u}{j}}\binom{u}{t}\). Let \(K\) be a \(t\)-clique chosen uniformly at random from all \(t\)-cliques of \(\mathcal{H}\), and let \((X_1, \dots, X_t)\) be a uniform random permutation of \(V(K)\).
For \(k\in[t]\), set \[x_k \mathrel{\vcenter{:}}= 2^{\mathbb{H}(X_k \mid X_1, \dots, X_{k-1})}.\] It follows from Propositions 3 and 4 that \[x_1 x_2 \cdots x_t = t! k_t^{(r)}(\mathcal{H}) > t! \frac{\binom{n}{j}}{\binom{u}{j}} \binom{u}{t} = j!\binom{n}{j} (u-j)(u-j-1)\cdots(u-t+1). \label{eq:total95product}\tag{1}\]
Since \((X_1, \dots, X_j)\) is an ordered \(j\)-tuple of distinct vertices in \(V\), the cardinality of its support is at most \(n!/(n-j)!\). By Proposition 3, we have \[x_1 \cdots x_j = 2^{\mathbb{H}(X_1, \dots, X_j)} \le n(n-1)\cdots(n-j+1) = j!\binom{n}{j}. \label{eq:prefix95bound}\tag{2}\] Dividing 1 by 2 , we obtain the tail bound \[\begin{align} \label{ine:tail95bound} x_{j+1} x_{j+2} \cdots x_t > (u-j)(u-j-1)\cdots(u-t+1). \end{align}\tag{3}\]
To decouple the constraints, we partition the sequence into three segments: the prefix \(A := x_1 \cdots x_j\), the extension \(B := x_{j+1} \cdots x_r\), and the tail \(C_{\mathrm{tail}} := x_{r+1} \cdots x_t\). The following claim shows that \(AB^p\) can be bounded in terms of the \(j\)-degrees of \(\mathcal{H}\).
Claim 5. We have \[AB^p \le j!(r - j)!^p \sum\limits_{s \in \binom{V}{j}} \deg(s)^p.\]
Proof. Let \(\vec{S} = (X_1, \dots, X_j)\) be the ordered prefix, and let \(S \in \binom{V}{j}\) be its underlying set. By Proposition 4, we have \[\begin{align} \label{EQ:another-form-ABp-THM} AB^p=2^{\mathbb{H}(\vec{S}) + p\mathbb{H}(X_{j+1},\dots,X_r | \vec{S})}. \end{align}\tag{4}\] Now, we bound \(\mathbb{H}(X_{j+1},\dots,X_r | \vec{S})\). Note that for a \(t\)-clique \(K\) in \(\mathcal{H}\), any \(r\)-subset of \(V(K)\) must be an edge in \(\mathcal{H}\). Consequently, for any valid prefix \(\vec{s} \in \text{supp}(\vec{S})\), the subsequent sequence \((X_{j+1}, \dots, X_r)\) together with \(s\) forms an \(r\)-edge in \(\mathcal{H}\), and the number of such extensions is at most \((r - j)!\deg(s)\). By Proposition 3, \[\begin{align} \mathbb{H}(X_{j+1}, \dots, X_r \mid \vec{S} = \vec{s}) \le \log_2((r - j)!\deg(s)). \end{align}\] Taking the expectation over all possible prefixes \(\vec{s} \in \text{supp}(\vec{S})\) yields that \[\begin{align} \mathbb{H}(X_{j+1}, \dots, X_r \mid \vec{S}) \le \sum_{\vec{s}\in \mathrm{supp}({\vec{S}})} \mathbb{P}(\vec{s}) \log_2((r - j)!\deg(s)). \end{align}\]
Recall the definition of entropy \(\mathbb{H}(\vec{S}) = \sum\limits_{\vec{s}\in \mathrm{supp}({\vec{S}})} \mathbb{P}(\vec{s}) \log_2 \frac{1}{\mathbb{P}(\vec{s})}\). Applying Jensen’s inequality to the concave function \(\log_2(x)\) yields \[\begin{align} \label{EQ:upper-bound-ABp-THM} \mathbb{H}(\vec{S}) + p\mathbb{H}(X_{j+1}, \dots, X_r \mid \vec{S}) &\le \sum_{\vec{s}\in \mathrm{supp}({\vec{S}})} \mathbb{P}(\vec{s}) \log_2 \left( \frac{((r - j)!\deg(s))^p}{\mathbb{P}(\vec{s})} \right) \notag\\ &\le \log_2 \left( \sum_{\vec{s}\in \mathrm{supp}({\vec{S}})} ((r - j)!\deg(s))^p \right). \end{align}\tag{5}\] For each underlying \(j\)-set \(s\) appearing in the support of \(S\), all \(j!\) orderings of \(s\) appear in \(\mathrm{supp}(\vec{S})\). Hence, grouping the terms by their underlying \(j\)-sets gives \[\label{EQ:underlying-support-THM} \sum_{\vec{s}\in \mathrm{supp}(\vec{S})} ((r-j)!\deg(s))^p = j!(r-j)!^p \sum_{s\in \mathrm{supp}(S)}\deg(s)^p.\tag{6}\] Since \(\deg(s)^p \ge 0\), extending this sum to all subsets in \(\binom{V}{j}\) can only increase the right-hand side. Combining 4 , 5 and 6 yields \[AB^p=2^{\mathbb{H}(\vec{S}) + p\mathbb{H}(X_{j+1},\dots,X_r | \vec{S})} \le j!(r - j)!^p \sum_{s \in \binom{V}{j}} \deg(s)^p,\] completing the proof of Claim 5. ◻
We derive a conflicting lower bound on \(AB^p\) by applying our continuous interpolation to the tail.
Claim 6. \(AB^p > j!\binom{n}{j}((u-j)\cdots(u-r+1))^p.\)
Proof. Rewriting the entropy product, it follows from 1 that \[\begin{align} \label{EQ:main-ineq-ABp-THM} AB^p = (ABC_{\text{tail}}) \frac{B^{p-1}}{C_{\text{tail}}} > \left[ j!\binom{n}{j} (u-j)\cdots(u-t+1) \right] \frac{B^{p-1}}{C_{\text{tail}}}. \end{align}\tag{7}\] To complete the proof, it suffices to give a lower bound for \(\frac{B^{p-1}}{C_{\text{tail}}}\).
Since entropy is non-negative, we naturally have \(x_k \ge 1\) for all \(k \in [t]\), which guarantees \(B \ge 1\). Consider the function \(f_j(y)=\prod_{i=j}^{r-1}(y - i)\). Clearly, \(f_j(y)\) is strictly increasing on the interval \([r - 1, \infty)\) with \(f_j(r-1)=0\). There exists a unique \(x' \in (r-1, \infty)\) such that \[\begin{align} \label{eq:B} B = \prod_{i=j}^{r-1}(x' - i). \end{align}\tag{8}\]
We claim that \[\begin{align} \label{EQ:bound-xr-with-x39-THM} x_r \le x' - r + 1. \end{align}\tag{9}\] Indeed, Lemma 1 gives that \[\begin{align} \label{EQ:gap-xk-xk431-THM} x_k - x_{k+1} \ge 1 \text{ for } 1\le k< t. \end{align}\tag{10}\] If \(x_r > x' - r + 1\), then by 10 , \(x_k \ge x_r + r - k > x' - k + 1\) for every \(j+1 \le k \le r\). Taking the product over these terms gives \[\begin{align} B = \prod_{k=j+1}^r x_k>\prod_{k=j+1}^r (x'-k+1)=\prod_{i=j}^{r-1}(x'-i), \end{align}\] which contradicts 8 .
Combining 9 and 10 , we have \(x_k\le x'-k+1\) for all \(r\le k\le t\). This together with \(x_t\ge 1\) yields that \(x'\ge t\) and then \[\begin{align} \label{ine:C95tail95bound} C_{\text{tail}}=x_{r+1}\cdots x_t \le \prod_{i=r}^{t-1}(x'-i). \end{align}\tag{11}\] By 3 , 8 and 11 , we conclude that \[\begin{align} \prod_{i=j}^{t-1}(u-i) < B C_{\text{tail}} \le \prod_{i=j}^{t-1}(x'-i). \end{align}\] Since the function \(g_j(y)=\prod_{i=j}^{t-1}(y-i)\) is strictly increasing on \((t-1,\infty)\) and \(x',u>t-1\), the above inequality implies that \(x'>u\ge \tilde{s}_R\). Since \(p>\frac{t-j}{r-j}>1\), using 8 and 11 , we obtain \[\begin{align} \frac{B^{p-1}}{C_{\text{tail}}} \ge \frac{((x'-j)\cdots(x'-r+1))^{p-1}}{(x'-r)\cdots(x'-t+1)} = \frac{j!(r-j)!^p}{t!} \cdot \frac{1}{\tilde{h}(x')}. \end{align}\]
Recall that \(\tilde{h}(x)\) is strictly decreasing on \([\widetilde{s}_R,\infty)\). It follows from \(x'>u\ge \widetilde{s}_R\) that \[\begin{align} \frac{B^{p-1}}{C_{\text{tail}}} > \frac{((u-j)\cdots(u-r+1))^{p-1}}{(u-r)\cdots(u-t+1)}. \end{align}\]
Substituting this into 7 establishes that \[AB^p > j!\binom{n}{j}((u-j)\cdots(u-r+1))^p,\] completing the proof of Claim 6. ◻
Combining Claims 5 and 6, we thus conclude that \[\begin{align} j!(r - j)!^p \sum_{s \in \binom{V}{j}} \deg(s)^p > j!\binom{n}{j}((u-j)\cdots(u-r+1))^p. \end{align}\] Dividing both sides by \(j!(r - j)!^p\), we obtain \[\begin{align} \sum_{s \in \binom{V}{j}} \deg(s)^p > \binom{n}{j} \left( \frac{(u-j)\cdots(u-r+1)}{(r-j)!} \right)^p = \binom{n}{j}\binom{u-j}{r-j}^p =C^p. \end{align}\]
This contradicts the norm constraint \(\sum\limits_{s \in \binom{V}{j}} \deg(s)^p \le C^p\). This completes the proof of Theorem 2.
In this section, we construct the extremal \(r\)-graph \(\mathcal{H}^*\) that exactly achieves the upper bound in Theorem 2. The extremal structure is a highly symmetric combinatorial design based on a Steiner system, which naturally unifies and generalizes the disjoint union of cliques known for the \(j = 1\) case.
Proof of Proposition 2. Fix \(j\) and \(u\), and let \(n\) be a sufficiently large integer satisfying ?? . By a result of Keevash [30], there exists a Steiner system \(S(j, u, n)=(V, \mathcal{B})\) such that each block \(B \in \mathcal{B}\) has size \(u\), and every \(j\)-subset \(S \in \binom{V}{j}\) is contained in exactly one block. We construct the \(r\)-graph \(\mathcal{H}^*\) on \(V\) by replacing each block in \((V, \mathcal{B})\) with a complete \(r\)-graph; that is, the edge set \(E^*\) of \(\mathcal{H}^*\) is \[\begin{align} E^* = \bigcup_{B \in \mathcal{B}} \binom{B}{r}. \end{align}\]
First, we verify the \((j, p)\)-norm of \(\mathcal{H}^*\). For any \(S \in \binom{V}{j}\), let \(B_S \in \mathcal{B}\) denote the unique block containing \(S\). By the definition of \(E^*\), the set of edges containing \(S\) is entirely restricted to \(B_S\): \[\begin{align} \{e \in E^* : S \subseteq e\} = \left\{e \in \binom{B_S}{r} : S \subseteq e \right\}. \end{align}\] Thus, the degree of every \(j\)-subset is exactly \(\deg_{\mathcal{H}^*}(S) = \binom{|B_S|-j}{r-j} = \binom{u-j}{r-j}\). Evaluating the norm yields \[\begin{align} \|\mathcal{H}^*\|_{j, p}^p = \sum_{S \in \binom{V}{j}} \deg_{\mathcal{H}^*}(S)^p = \sum_{S \in \binom{V}{j}} \binom{u-j}{r-j}^p = \binom{n}{j}\binom{u-j}{r-j}^p. \end{align}\]
Next, we evaluate the number of \(t\)-cliques in \(\mathcal{H}^*\) by showing that for every \(t\)-clique \(K\) in \(\mathcal{H}^*\), there exists a unique block \(B \in \mathcal{B}\) such that \(K \subseteq B\).Suppose for contradiction that there exists a \(t\)-clique \(K\) such that \(K \not\subseteq B\) for all \(B \in \mathcal{B}\). Fix \(S \in \binom{K}{j}\), and let \(B_a \in \mathcal{B}\) be the unique block containing \(S\). Since \(K \not\subseteq B_a\), we can choose \(v \in K \setminus B_a\). Given \(t \ge r \ge j+1\), there exists an \(r\)-subset \(e\) satisfying \[\begin{align} S \cup \{v\} \subseteq e \subseteq K. \end{align}\] Since \(K\) is a clique, \(e \in E^*\). By the construction of \(E^*\) and the Steiner property, there exists a unique block \(B_b \in \mathcal{B}\) such that \(e \subseteq B_b\). This establishes the following chain of inclusions \[\begin{align} S \subseteq S \cup \{v\} \subseteq e \subseteq B_b. \end{align}\] However, \(S \subseteq B_b\) uniquely determines \(B_b = B_a\). This implies \(v \in e \subseteq B_a\), directly contradicting \(v \in K \setminus B_a\).
Consequently, the set of \(t\)-cliques in \(\mathcal{H}^*\) is partitioned by the blocks in \(\mathcal{B}\). Since \(|\mathcal{B}| = \binom{n}{j}/\binom{u}{j}\) , we have \[k_t^{(r)}(\mathcal{H}^*) = \sum_{B \in \mathcal{B}} k_t^{(r)}(B) = |\mathcal{B}|\binom{u}{t} = \frac{\binom{n}{j}}{\binom{u}{j}}\binom{u}{t}.\] This completes the proof of Proposition 2. ◻
In this note, we answer the fixed-vertex question raised by Chao–Dong–Shen–Yang in the remaining range \[p>\frac{t-j}{r-j} \quad\text{and}\quad C^p \ge \binom{n}{j}\binom{\widetilde{s}_{\mathbb{R}}-j}{r-j}^p .\] The proof isolates the new difficulty caused by the fixed vertex set in the hypergraph setting: the entropy of the initial \(j\)-set must be coupled with the local \(j\)-degree norm. This is handled by splitting a random ordered clique into a \(j\)-prefix, an \((r-j)\)-extension, and a tail, and by applying a continuous interpolation step.
The upper bound is sharp whenever the corresponding Steiner systems exist. It would be interesting to understand the exact extremal configurations in the non-divisible cases, and to establish corresponding stability results. Another possible direction is to study analogous counting problems for other fixed hypergraphs under \((j,p)\)-norm constraints.