May 23, 2026
We revisit a conjecture of Mubayi that was proposed as a hypergraph analogue of the Li–Li algebraic proof of Turán’s theorem. The conjecture compares a polynomial ideal generated by multipartite 3-graphs with a differentiated diagonal-vanishing ideal. We show that the proposed equality fails for every non-vacuous choice of parameters. The obstruction is structural: diagonal vanishing does not remember the missing codegree-star condition that drives Mubayi’s hypergraph problem.
We then give a replacement in edge-variable rings using monomial cover ideals. For ordinary forbidden-family Turán problems, the cover ideal converts extremal edge counting into an initial-degree computation. For generalized Turán numbers, the same cover ideal encodes the forbidden condition, while the objective becomes a quotient rank on the space spanned by the target-copy monomials.
For Mubayi’s core-pair family \(\mathcal{K}_{\ell}^{(r)}\), this cover ideal has an explicit missing codegree-star form. A Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients computes its initial degree and recovers Mubayi’s hypergraph Turán theorem.
Keywords. Turán problems; codegree-star ideals; cover ideals; Hilbert functions.
Given a family \(\mathcal{F}\) of \(r\)-uniform hypergraphs, or \(r\)-graphs for short, and an \(r\)-graph \(G\), we say that \(G\) is \(\mathcal{F}\)-free if it contains no member of \(\mathcal{F}\) as a subhypergraph. The extremal number \(\operatorname{ex}(n,\mathcal{F})\) is the maximum number of edges in an \(\mathcal{F}\)-free \(r\)-graph on \(n\) vertices. In the special case that \(\mathcal{F}=\{F\}\), we write \(\operatorname{ex}(n,F)\).
The case \(r=2\) is the classical starting point of extremal graph theory. Turán’s theorem determines \(\operatorname{ex}(n,K_{q+1})\) exactly [1], and the Erdős–Stone theorem gives the asymptotic answer for every non-bipartite forbidden graph [2]. Variants in which one maximizes the number of copies of a fixed graph also go back to Zykov’s symmetrization and clique-counting theorem [3]; this viewpoint is now part of the generalized Turán problem studied systematically by Alon and Shikhelman [4]. Hypergraph Turán problems are much less rigid, and even natural analogues of complete graphs often require additional structure.
One important source of such analogues is the theory of graph expansions. Given a graph \(F\), its \(r\)-uniform expansion is obtained by enlarging each edge of \(F\) with new vertices, using disjoint new vertex sets for distinct edges. The expansion of a clique is a particularly natural hypergraph substitute for a complete graph. Mubayi’s family \(\mathcal{K}_{\ell}^{(r)}\) contains this expanded clique as a canonical member, and his theorem for the whole core-pair family gives a robust family version of the corresponding Turán phenomenon [5]. Stability aspects of this extension were later revisited by Liu [6]. Pikhurko later obtained the exact Turán number for expanded complete graphs for all sufficiently large \(n\) [7]. More broadly, Turán problems for expansions have developed into an active topic; see, for example, the survey of Mubayi and Verstraëte [8] and the work of Kostochka, Mubayi, and Verstraëte on expansions and shadows [9].
One algebraic approach to Turán’s theorem is due to Li and Li [10]. For a graph \(G\) on vertex set \([n]\), let \(p_G(x_1,\ldots,x_n)=\prod_{ij\notin G}(x_i-x_j)\). The absence of a clique can be detected by membership in an ideal of polynomials vanishing after variables are identified. Since \(\deg p_G\) is the number of missing edges, this gives an algebraic proof of the Turán bound.
Mubayi observed in [5] that this proof does not immediately extend to hypergraphs. To capture the condition that a pair of vertices has codegree zero in a 3-graph, he proposed a differentiated diagonal-vanishing ideal. For a 3-graph \(G\) on \([n]\), set \[p_G(x_1,\ldots,x_n)=\prod_{ijk\notin G}(x_i-x_j)(x_i-x_k)(x_j-x_k).\] Let \(DI(n,\ell)\) be the ideal of polynomials whose relevant partial derivatives vanish whenever any \(\ell\) variables are identified, and let \(\widehat P^{(3)}(n,\ell)\) be the ideal generated by the polynomials \(p_G\) with \(G\) ranging over all \((\ell-1)\)-partite 3-graphs. Mubayi’s Conjecture 6 asserts that \(\widehat P^{(3)}(n,\ell)=DI(n,\ell)\). An affirmative answer would imply the hypergraph Turán upper bound for the family \(\mathcal{K}_{\ell}^{(3)}\) introduced below.
Our first result shows that this conjectural equality is false in the strongest possible sense.
Theorem 1. For every \(\ell\ge 3\) and every \(n\ge \ell\), \(\widehat P^{(3)}(n,\ell)\subsetneq DI(n,\ell)\). When \(n<\ell\), both ideals are the whole polynomial ring. Thus the strict containment holds precisely in the non-vacuous range.
The counterexamples are elementary Vandermonde-type products. They show that diagonal vanishing can be distributed among many pair factors without forcing an entire missing codegree star. This is the basic reason that \(DI(n,\ell)\) is too large for the intended hypergraph application.
The second part of the paper gives an algebraic replacement that retains the relevant combinatorial information. We work in an edge-variable ring \(R_{n,r}=\Bbbk[y_E:E\in \binom{[n]}r]\), one variable for each possible hyperedge. The forbidden copies define a Stanley–Reisner ideal, and the Alexander dual is the corresponding cover ideal. This language is standard in combinatorial commutative algebra [11]–[14]; here it is used as a direct dictionary between missing edges and monomial divisibility.
For ordinary extremal problems, the dictionary is exact: \(\alpha(C_{\mathcal{F},n})=\binom nr-\operatorname{ex}(n,\mathcal{F})\). For generalized Turán numbers in the sense of Alon and Shikhelman [4], where one forbids a graph \(F\) and maximizes the number of copies of another graph \(T\), ordinary degree must be replaced by a quotient-theoretic rank on the vector space spanned by the \(T\)-copy monomials. This gives \(\operatorname{ex}(n,T,F)=|\mathcal{T}_{T,n}|-\alpha_T(C_{\mathcal{F},n})\), where \(\alpha_T\) is the minimum dimension of the part of this target-copy space killed by a missing-edge quotient. When \(T=K_2\), this is the usual initial degree.
Finally, we apply the method to Mubayi’s core-pair family. Let \(\mathcal{K}_{\ell}^{(r)}\) be the family of \(r\)-graphs with at most \(\binom{\ell}{2}\) edges and with a distinguished \(\ell\)-vertex core such that every pair of core vertices lies in an edge. The cover ideal of this forbidden family is not arbitrary: it is exactly a missing codegree-star ideal. Its initial degree is computed below by translating its monomials into square-zero quadratic monomial quotients and applying a Hilbert-function symmetrization theorem, an algebraic form of the same smoothing principle behind Zykov’s theorem. This yields \(\operatorname{ex}(n,\mathcal{K}_{\ell}^{(r)})=t_r(n,\ell-1)\), where \(t_r(n,q)\) is the number of edges in the complete balanced \(q\)-partite \(r\)-graph on \(n\) vertices. This recovers Mubayi’s hypergraph Turán theorem in a form adapted to monomial ideals.
Throughout, \([n]=\{1,\ldots,n\}\) and \(\Bbbk\) is a field of characteristic zero. We identify a hypergraph with its edge set whenever no confusion can arise. If \(G\) is a hypergraph, then \(V(G)\) denotes its vertex set. Given an \(r\)-graph \(G\) and vertices \(x,y\in V(G)\), the codegree \(\operatorname{codeg}_G(x,y)\) is the number of edges of \(G\) containing both \(x\) and \(y\).
For positive integers \(n,q,r\), let \(T_r(n,q)\) denote the complete balanced \(q\)-partite \(r\)-graph on \(n\) vertices. Thus the vertex set is partitioned into \(q\) classes, no two class sizes differ by more than one, and the edges are all \(r\)-sets meeting each class in at most one vertex. Let \(t_r(n,q)=|T_r(n,q)|\). Equivalently, if the balanced class sizes are \(n_1,\ldots,n_q\), then \[t_r(n,q)=\sum_{S\in\binom{[q]}r}\prod_{i\in S}n_i,\] with the convention that \(t_r(n,q)=0\) when \(r>q\).
Definition 1 (Mubayi’s family \(\mathcal{K}_{\ell}^{(r)}\)). Fix \(\ell,r\ge 2\). Let \(\mathcal{K}_{\ell}^{(r)}\) be the family of all \(r\)-graphs with at most \(\binom{\ell}{2}\) edges which contain a set \(S\) of \(\ell\) vertices, called the core, such that every pair of vertices in \(S\) is contained in at least one edge.
When \(r=2\), this family consists of the complete graph \(K_\ell\). For \(r>2\), the same definition encodes a positive codegree condition on all pairs in the core.
Let \(S_n=\Bbbk[x_1,\ldots,x_n]\). For a 3-graph \(G\) with vertex set contained in \([n]\), define \[p_G=\prod_{\substack{1\le i<j<k\le n\\ ijk\notin G}}(x_i-x_j)(x_i-x_k)(x_j-x_k).\] Let \(I(n,\ell)\subset S_n\) be the ideal of polynomials that vanish after the identification of any \(\ell\) variables. Following Mubayi, define \[DI(n,\ell)=\left\{p\in S_n: \frac{\partial^j p}{\partial x_i^j}\in I(n,\ell)\text{ for every }i\in[n]\text{ and every }0\le j\le n-3 \right\}.\] Let \(T_{\ell-1}^{(3)}\) be the family of all \((\ell-1)\)-partite 3-graphs with vertex set contained in \([n]\), and put \(\widehat P^{(3)}(n,\ell)=\bigl(p_G:G\in T_{\ell-1}^{(3)}\bigr)\subset S_n\). Mubayi’s Conjecture 6 is the proposed equality \(\widehat P^{(3)}(n,\ell)=DI(n,\ell)\).
We now prove Theorem 1. We first record the inclusion direction; the counterexamples below show that the reverse inclusion fails.
Lemma 1. For all \(\ell\ge 3\) and \(n\ge \ell\), \(\widehat P^{(3)}(n,\ell)\subseteq DI(n,\ell)\).
Proof. Since \(DI(n,\ell)\) is an ideal, it is enough to show that each generator \(p_G\) belongs to \(DI(n,\ell)\), where \(G\) is an \((\ell-1)\)-partite 3-graph on \([n]\). Fix an \(\ell\)-set \(L\subset[n]\). By the pigeonhole principle, two vertices \(a,b\in L\) lie in the same part of the \((\ell-1)\)-partition. No edge of \(G\) contains both \(a\) and \(b\). Hence every triple \(\{a,b,c\}\) with \(c\in[n]\setminus\{a,b\}\) is missing from \(G\), and the factor \((x_a-x_b)\) occurs once for each such \(c\). Thus \((x_a-x_b)^{n-2}\mid p_G\). Now fix \(i\in[n]\) and \(0\le j\le n-3\). If \(i\notin\{a,b\}\), this factor is independent of \(x_i\) and divides \(\partial^j p_G/\partial x_i^j\). If \(i=a\) or \(i=b\), differentiating \(j\) times can reduce the exponent of \((x_a-x_b)\) by at most \(j\), so every term of \(\partial^j p_G/\partial x_i^j\) remains divisible by \((x_a-x_b)^{n-2-j}\), which has positive exponent. Therefore \(\partial^j p_G/\partial x_i^j\) vanishes when the variables indexed by \(L\) are identified. Since \(L\), \(i\), and \(j\) were arbitrary, \(p_G\in DI(n,\ell)\). ◻
The following statement records the full parameter range.
Theorem 2. Let \(\ell\ge 3\) and \(n\ge \ell\). Then \(\widehat P^{(3)}(n,\ell)\subsetneq DI(n,\ell)\). Consequently, Mubayi’s polynomial-ideal conjecture fails for every non-vacuous pair of parameters.
Proof. By Lemma 1, it remains only to exhibit an element of \(DI(n,\ell)\) which is not in \(\widehat P^{(3)}(n,\ell)\). We split the construction into three cases.
Case 1: \(\ell\ge 4\). Put \(q=\ell-1\), so \(q\ge 3\), and choose a balanced partition \([n]=V_1\sqcup\cdots\sqcup V_{\ell-2}\) into \(\ell-2=q-1\) nonempty parts. Define the Vandermonde-type product \[F_{n,\ell}=\prod_{s=1}^{\ell-2}\prod_{\substack{a<b\\ a,b\in V_s}}(x_a-x_b).\] We first verify that \(F_{n,\ell}\in DI(n,\ell)\). Fix an \(\ell\)-set \(L\subset[n]\) and a variable \(x_i\). If \(i\in L\), then the \(\ell-1\) vertices of \(L\setminus\{i\}\) are distributed among only \(\ell-2\) parts, and hence two of them, say \(a\) and \(b\), lie in the same part. The factor \(x_a-x_b\) divides \(F_{n,\ell}\) and is independent of \(x_i\). Therefore it also divides every derivative \(\partial^jF_{n,\ell}/\partial x_i^j\).
If \(i\notin L\), then the \(\ell\) vertices of \(L\) themselves are distributed among \(\ell-2\) parts, so the same argument gives a pair \(a,b\in L\) with \(x_a-x_b\) dividing every derivative with respect to \(x_i\). In both cases the derivative vanishes on the diagonal indexed by \(L\). Since \(L,i\), and \(j\) were arbitrary, \(F_{n,\ell}\in DI(n,\ell)\).
It remains to show that \(F_{n,\ell}\notin \widehat P^{(3)}(n,\ell)\). Each generator \(p_G\) of \(\widehat P^{(3)}(n,\ell)\) is homogeneous of degree \(\deg p_G=3\left(\binom n3-|G|\right)\). The largest number of edges in an \((\ell-1)\)-partite 3-graph on \([n]\) is \(t_3(n,\ell-1)=t_3(n,q)\). Hence all homogeneous generators of \(\widehat P^{(3)}(n,\ell)\) have degree at least \(D_{n,\ell}=3\left(\binom n3-t_3(n,q)\right)\). Since \(\widehat P^{(3)}(n,\ell)\) is homogeneous and generated in degrees at least \(D_{n,\ell}\), it contains no nonzero homogeneous element of smaller degree.
We now compare degrees. The chosen partition has \(q-1\) balanced classes, and therefore \[\deg F_{n,\ell}=\sum_{s=1}^{q-1}\binom{|V_s|}{2}<\frac{n^2}{2(q-1)}.\] Let \(W_1,\ldots,W_q\) be the balanced \(q\)-partition defining \(T_3(n,q)\), and set \(a=\left\lceil n/q\right\rceil\). The triples missing from \(T_3(n,q)\) include all triples having two vertices in one part and the third vertex outside that part. Thus \[\binom n3-t_3(n,q)\ge \sum_{s=1}^q\binom{|W_s|}{2}(n-|W_s|).\] Since \(|W_s|\le a\) for every \(s\) and \(n\ge q+1\), we have \(\sum_{s=1}^q\binom{|W_s|}{2}\ge n-q\), and consequently \(\binom n3-t_3(n,q)\ge (n-q)(n-a)\). It follows that \(D_{n,\ell}\ge 3(n-q)(n-a)\). We claim that \(3(n-q)(n-a)>n^2/(2(q-1))\). Indeed, \(a\le (n+q-1)/q\), whence \(n-a\ge (q-1)(n-1)/q\). Moreover, the function \((n-q)(n-1)/n^2\) is increasing for \(n\ge q+1\) and equals \(q/(q+1)^2\) at \(n=q+1\). Therefore \[\frac{6(q-1)(n-q)(n-a)}{n^2} \ge \frac{6(q-1)^2}{q}\cdot \frac{(n-q)(n-1)}{n^2} \ge \frac{6(q-1)^2}{(q+1)^2}>1\] for every \(q\ge 3\). Hence \(\deg F_{n,\ell}<D_{n,\ell}\), which proves \(F_{n,\ell}\notin \widehat P^{(3)}(n,\ell)\).
Case 2: \(\ell=3\) and \(n\ge 4\). Take the full Vandermonde product \(F_{n,3}=\prod_{1\le a<b\le n}(x_a-x_b)\). For any 3-set \(L\) and any \(i\in[n]\), if \(i\in L\) then the two vertices of \(L\setminus\{i\}\) give a factor independent of \(x_i\); if \(i\notin L\), any two vertices of \(L\) give such a factor. Thus every derivative with respect to \(x_i\) that appears in the definition of \(DI(n,3)\) vanishes on the diagonal indexed by \(L\). Hence \(F_{n,3}\in DI(n,3)\).
A 2-partite 3-graph has no edges. Therefore every generator of \(\widehat P^{(3)}(n,3)\) has degree \(3\binom n3\). On the other hand, \(\deg F_{n,3}=\binom n2<3\binom n3\) for \(n\ge 4\). Hence \(F_{n,3}\notin \widehat P^{(3)}(n,3)\).
Case 3: \((\ell,n)=(3,3)\). Here \(n-3=0\), so \(DI(3,3)=I(3,3)\). The linear polynomial \(F_{3,3}=x_1-x_2\) vanishes on the diagonal \(x_1=x_2=x_3\), and therefore belongs to \(DI(3,3)\). However, every 2-partite 3-graph on \([3]\) has no edge, so \(\widehat P^{(3)}(3,3)\) is generated by \((x_1-x_2)(x_1-x_3)(x_2-x_3)\). This ideal contains no nonzero element of degree one. Hence \(F_{3,3}\notin \widehat P^{(3)}(3,3)\).
The three cases prove the theorem. ◻
Remark 3. When \(n<\ell\), the defining condition for \(I(n,\ell)\) is vacuous, so \(DI(n,\ell)=S_n\). In the same range, the complete 3-graph on \([n]\) is \((\ell-1)\)-partite, and the corresponding generator is \(p_G=1\). Thus \(\widehat P^{(3)}(n,\ell)=S_n\) as well. Theorem 2 therefore covers exactly the range in which the conjecture has content.
Remark 4. The construction explains the failure mechanism. The polynomial \(F_{n,\ell}\) has enough diagonal vanishing because every \(\ell\)-set contains an internal pair in a fixed \((\ell-2)\)-partition. This is weaker than forcing a complete missing codegree star. The monomial ideals introduced below retain precisely this missing-star information.
This section supplies the algebraic extremal input used later in place of an auxiliary-graph clique-counting theorem. It will be applied twice: first to the clique-counting generalized Turán problem, and then to the initial-degree computation for the codegree-star ideal. The objects are quadratic monomial quotients with all variables square-zero. Their Hilbert functions encode the same numerical constraints, but the proof below is phrased entirely in terms of standard monomials, links, quotients, and cloning of variables.
Let \(S=\Bbbk[x_1,\ldots,x_n]\), and let \(A=S/I\) be a quotient of the form \[I=(x_1^2,\ldots,x_n^2,\;x_ax_b\text{ for some pairs }\{a,b\}\subset[n]).\] We call such an \(A\) a square-zero quadratic monomial quotient. A squarefree monomial \(x_{i_1}\cdots x_{i_d}\) is called standard if its image in \(A_d\) is nonzero. The standard monomials form a \(\Bbbk\)-basis of \(A\).
For fixed \(q\ge 1\), define \[\begin{align} \mathcal{H}(n,q,r)=\max\{\,&\dim_\Bbbk A_r: A\text{ is a square-zero quadratic monomial quotient on }n\text{ variables,}\\ & A_{q+1}=0\}. \end{align}\] The main result of this section is \(\mathcal{H}(n,q,r)=t_r(n,q)\). The proof below gives the upper bound, which is the only direction needed later. The reverse inequality is attained by the complete balanced \(q\)-partite square-zero quotient: take a balanced partition of \([n]\) into \(q\) parts and kill exactly the squares and the products of variables in the same part. Its degree-\(r\) Hilbert function is \(t_r(n,q)\) and its degree-\((q+1)\) piece is zero.
Two variables \(x_i,x_j\) are called parallel in \(A\) if \(x_ix_j=0\) and, for every \(h\notin\{i,j\}\), \(x_ix_h=0\) if and only if \(x_jx_h=0\). This is an equivalence relation on the variables. Its equivalence classes will be called parallel classes. Inside each parallel class all pairwise products vanish; between two distinct classes either all products vanish or all products are nonzero.
If \(C\) is a parallel class and \(x_c\in C\), set \[\lambda_C(d)=\dim_\Bbbk\operatorname{span}\{w\in A_d:x_cw\ne 0\text{ in }A\}.\] Equivalently, \(\lambda_C(d)=\dim_\Bbbk(x_cA_d)\). This number is independent of the choice of \(x_c\in C\).
Lemma 2 (Algebraic cloning). Let \(A\) be a square-zero quadratic monomial quotient with \(A_{q+1}=0\). Let \(U\) and \(V\) be distinct parallel classes such that \(x_ux_v=0\) for all \(u\in U\) and \(v\in V\). Fix a degree \(r\). Then one can replace either \(U\) by clones of \(V\) or \(V\) by clones of \(U\) so as to obtain another square-zero quadratic monomial quotient \(A'\) on the same \(n\) variables with \((A')_{q+1}=0\) and \(\dim_\Bbbk(A')_r\ge \dim_\Bbbk A_r\). Moreover, this replacement merges the two classes into one parallel class and leaves all other products unchanged except those involving the cloned class.
Proof. We describe the operation \(V\leftarrow U\). Let \(A^{V\leftarrow U}\) be the quotient obtained as follows. All products inside \(U\cup V\) are set equal to zero. For \(v\in V\) and for a variable \(z\notin U\cup V\), we impose \(x_vz=0\) if and only if \(x_uz=0\) for some (equivalently, every) \(u\in U\). All products not involving \(V\) are kept as in \(A\). Thus the variables in \(V\) become clones of the variables in \(U\).
First, \(A^{V\leftarrow U}_{q+1}=0\). Indeed, a standard monomial of degree \(q+1\) in \(A^{V\leftarrow U}\) that avoids \(V\) would already have been standard in \(A\), impossible. A standard monomial that contains a variable \(x_v\) with \(v\in V\) contains no other variable from \(U\cup V\); after replacing \(x_v\) by any \(x_u\) with \(u\in U\), its support would be standard in \(A\) by construction. This would give a nonzero element of \(A_{q+1}\), again impossible.
Now compare degree \(r\). The standard monomials of degree \(r\) avoiding \(V\) are unchanged. Since products inside \(V\) and between \(U\) and \(V\) are zero before and after cloning, a standard degree-\(r\) monomial containing a variable of \(V\) contains exactly one such variable and no variable from \(U\). In \(A\) these monomials are counted by \(|V|\lambda_V(r-1)\), whereas in \(A^{V\leftarrow U}\) they are counted by \(|V|\lambda_U(r-1)\). Consequently \[\dim_\Bbbk A^{V\leftarrow U}_r-\dim_\Bbbk A_r =|V|\bigl(\lambda_U(r-1)-\lambda_V(r-1)\bigr).\] If this quantity is nonnegative, take \(A'=A^{V\leftarrow U}\). Otherwise perform the opposite cloning \(U\leftarrow V\), for which the corresponding difference is \(|U|\bigl(\lambda_V(r-1)-\lambda_U(r-1)\bigr)>0\). In either case the degree-\(r\) Hilbert function does not decrease, and the classes \(U,V\) are merged. ◻
Theorem 5 (Hilbert-function Turán bound). Let \(A=\Bbbk[x_1,\ldots,x_n]/I\) be a square-zero quadratic monomial quotient. If \(A_{q+1}=0\), then, for every \(r\ge 0\), \(\dim_\Bbbk A_r\le t_r(n,q)\). Here \(t_r(n,q)\) is the degree-\(r\) Hilbert function of the complete balanced \(q\)-partite square-zero quotient. The bound is sharp.
Proof. The cases \(r=0\) and \(r=1\) are immediate, so fix \(r\ge 2\). Starting from \(A\), repeatedly apply Lemma 2 whenever two distinct parallel classes have zero product between them. Each step preserves the condition that the degree-\((q+1)\) piece is zero and does not decrease the degree-\(r\) Hilbert function. It also decreases the number of parallel classes. Hence the process terminates after finitely many steps.
Let \(B\) be the terminal quotient. Then \(\dim_\Bbbk A_r\le \dim_\Bbbk B_r\) and \(B_{q+1}=0\), and any two distinct parallel classes of \(B\) have nonzero product between them. Thus there is a partition \([n]=P_1\sqcup\cdots\sqcup P_s\) such that \(x_ix_j=0\) if and only if \(i,j\) belong to the same \(P_a\). Writing \(n_a=|P_a|\), the standard degree-\(r\) monomials of \(B\) are obtained by choosing \(r\) distinct classes and one variable from each chosen class. Hence \(\dim_\Bbbk B_r=e_r(n_1,\ldots,n_s)\), where \(e_r\) denotes the elementary symmetric polynomial of degree \(r\).
Since \(B_{q+1}=0\), one must have \(s\le q\); otherwise a product of one variable from each of \(q+1\) distinct classes would be nonzero. By appending zero parts, we may regard \((n_1,\ldots,n_s)\) as a \(q\)-tuple summing to \(n\).
It remains to show that \(e_r(n_1, \ldots,n_q)\) is maximized by a balanced \(q\)-tuple. Suppose two entries satisfy \(a\ge b+2\), and let \(\mathbf{c}\) denote the remaining entries. Replacing \((a,b)\) by \((a-1,b+1)\) changes the value of \(e_r\) by \[e_r(\mathbf{c},a-1,b+1)-e_r(\mathbf{c},a,b) =(a-b-1)e_{r-2}(\mathbf{c})\ge 0,\] with the convention \(e_0=1\) and \(e_d=0\) for \(d<0\). Repeated smoothing therefore leads to a balanced \(q\)-tuple without decreasing \(e_r\). Thus \(\dim_\Bbbk B_r\le t_r(n,q)\), and the theorem follows. ◻
Remark 6. Theorem 5 is the form of the Zykov clique-counting theorem needed in the sequel [3]. Applied to the Stanley–Reisner quotient of a graph \(F\), the number \(\dim_\Bbbk A_r\) counts \(r\)-cliques of \(F\), and the condition \(A_{q+1}=0\) says that \(F\) has no \((q+1)\)-clique. The proof above, however, does not use graph deletion, graph neighborhoods, or Zykov symmetrization; the symmetrization is carried out directly on the quadratic monomial quotient.
We now give the general algebraic dictionary. Let \(\mathcal{F}\) be a family of \(r\)-graphs, and let \(R_{n,r}=\Bbbk[y_E:E\in\binom{[n]}r]\) be the polynomial ring with one variable for each possible \(r\)-edge. Denote by \(\mathcal{C}_{\mathcal{F},n}\) the set of all copies, inside \(\binom{[n]}r\), of members of \(\mathcal{F}\).
The \(\mathcal{F}\)-free edge sets form a simplicial complex \[\Delta_{\mathcal{F},n}=\{G\subseteq \binom{[n]}r:G\text{ is }\mathcal{F}\text{-free}\}.\] Its Stanley–Reisner ideal is \[I_{\mathcal{F},n}=\left(\prod_{E\in H}y_E:H\in \mathcal{C}_{\mathcal{F},n}\right).\] The Alexander-dual cover ideal is \[C_{\mathcal{F},n}=I_{\mathcal{F},n}^{\vee} =\bigcap_{H\in \mathcal{C}_{\mathcal{F},n}}(y_E:E\in H).\] Thus a monomial lies in \(C_{\mathcal{F},n}\) precisely when its support meets every forbidden copy.
Theorem 7 (Cover-ideal Turán dictionary). For every family \(\mathcal{F}\) of \(r\)-graphs, \(\alpha(C_{\mathcal{F},n})=\binom nr-\operatorname{ex}(n,\mathcal{F})\). Moreover, the squarefree monomials of degree \(\alpha(C_{\mathcal{F},n})\) are exactly the complements of extremal \(\mathcal{F}\)-free \(r\)-graphs.
Proof. Since \(C_{\mathcal{F},n}\) is a squarefree monomial ideal, its initial degree is attained by a squarefree monomial. For \(M\subseteq \binom{[n]}r\), write \(y_M=\prod_{E\in M}y_E\). Then \(y_M\in C_{\mathcal{F},n}\) if and only if \(M\) intersects every forbidden copy \(H\in\mathcal{C}_{\mathcal{F},n}\). This is equivalent to saying that the complementary edge set \(\binom{[n]}r\setminus M\) is \(\mathcal{F}\)-free. Minimizing \(|M|\) over all squarefree monomials in \(C_{\mathcal{F},n}\) is therefore the same as maximizing the number of edges in an \(\mathcal{F}\)-free \(r\)-graph. Hence \(\alpha(C_{\mathcal{F},n})=\binom nr-\operatorname{ex}(n,\mathcal{F})\). The characterization of the monomials attaining the initial degree follows from the same equivalence. ◻
Remark 8. Theorem 7 is a dictionary rather than a solution theorem. For a general forbidden family, computing the initial degree of \(C_{\mathcal{F},n}\) is essentially the original extremal problem. The point is that some families, including Mubayi’s \(\mathcal{K}_{\ell}^{(r)}\), give cover ideals with additional structure.
Let \(T\) be the graph to be counted and \(F\) the graph to be forbidden. We assume, as usual, that both graphs have no isolated vertices. Let \(R_n^{(2)}=\Bbbk[y_e:e\in E(K_n)]\) be the edge-variable ring of the complete graph. Let \(\mathcal{T}_{T,n}\) be the set of edge sets of all copies of \(T\) in \(K_n\), and let \(\mathcal{C}_{F,n}\) be the set of edge sets of all copies of \(F\) in \(K_n\). The forbidden-copy ideal and its Alexander dual are \[I_{F,n}=(y_A:A\in \mathcal{C}_{F,n}), \qquad C_{F,n}=I_{F,n}^{\vee}=\bigcap_{A\in \mathcal{C}_{F,n}}(y_e:e\in A),\] where \(y_A=\prod_{e\in A}y_e\).
The target is encoded by the finite-dimensional monomial subspace \[V_T(n)=\operatorname{span}_\Bbbk\{y_B:B\in\mathcal{T}_{T,n}\}\subset R_n^{(2)}.\] For \(M\subseteq E(K_n)\), set \[\mathfrak m_M=(y_e:e\in M), \qquad Q_M=R_n^{(2)}/\mathfrak m_M, \qquad \pi_M:R_n^{(2)}\longrightarrow Q_M.\] The quotient rank of \(M\) on the target-copy space is \(\rho_T(M)=\dim_\Bbbk\pi_M(V_T(n))\). This is the number of target-copy monomials which remain nonzero after the variables in \(M\) are killed. Dually, define the \(T\)-initial defect of the cover ideal by \[\alpha_T(C_{F,n}) = \min_{\substack{y_M\in C_{F,n}\\ y_M\text{ squarefree}}} \dim_\Bbbk\ker\bigl(\pi_M|_{V_T(n)}\bigr) = \min_{\substack{y_M\in C_{F,n}\\ y_M\text{ squarefree}}} \dim_\Bbbk\bigl(V_T(n)\cap\mathfrak m_M\bigr).\] Thus \(\alpha_T\) is defined by a quotient map rather than by ordinary degree. Since \(V_T(n)\) has the monomial basis indexed by \(T\)-copies, this defect is equivalently the number of \(T\)-copy monomials killed by the missing-edge ideal \(\mathfrak m_M\).
Theorem 9 (Cover ideals for generalized Turán numbers). For fixed graphs \(T\) and \(F\) without isolated vertices, \(\operatorname{ex}(n,T,F)=|\mathcal{T}_{T,n}|-\alpha_T(C_{F,n})\). Equivalently, \[\operatorname{ex}(n,T,F)=\max_{\substack{y_M\in C_{F,n}\\ y_M\text{ squarefree}}}\dim_\Bbbk\pi_M(V_T(n)).\]
Proof. The squarefree monomials in \(C_{F,n}\) are exactly the monomial vertex covers of the family of forbidden-copy monomials \(\{y_A:A\in\mathcal{C}_{F,n}\}\). Equivalently, \(y_M\in C_{F,n}\) if and only if every forbidden-copy monomial \(y_A\) is killed in the quotient \(Q_M=R_n^{(2)}/\mathfrak m_M\).
For such an \(M\), the quotient \(Q_M\) has the standard monomial basis consisting of monomials in variables \(y_e\) with \(e\notin M\). Hence a basis element \(y_B\in V_T(n)\) has nonzero image under \(\pi_M\) precisely when none of its variables is killed, i.e.precisely when \(y_B\notin \mathfrak m_M\). Distinct nonzero images remain distinct standard monomials in \(Q_M\). Therefore \(\rho_T(M)=\dim_\Bbbk\pi_M(V_T(n))\) is exactly the number of target-copy monomials surviving in the quotient.
Thus maximizing the number of \(T\)-copies in an \(F\)-free graph is the same algebraic problem as maximizing \(\rho_T(M)\) over the squarefree monomials \(y_M\in C_{F,n}\). Since \(\dim_\Bbbk V_T(n)=|\mathcal{T}_{T,n}|\) and \[\dim_\Bbbk\pi_M(V_T(n)) = \dim_\Bbbk V_T(n)-\dim_\Bbbk\ker(\pi_M|_{V_T(n)}),\] the maximum quotient rank equals \[|\mathcal{T}_{T,n}|- \min_{\substack{y_M\in C_{F,n}\\ y_M\text{ squarefree}}} \dim_\Bbbk\ker(\pi_M|_{V_T(n)}).\] This is the asserted identity. ◻
Remark 10. When \(T=K_2\), the space \(V_T(n)\) is spanned by the edge variables themselves. For a squarefree monomial \(y_M\), the kernel of \(\pi_M\) on \(V_T(n)\) is the span of the variables \(y_e\) with \(e\in M\). Hence \(\alpha_{K_2}(C_{F,n})=\alpha(C_{F,n})\), and Theorem 9 reduces to Theorem 7. For general \(T\), ordinary degree no longer measures the objective; the relevant invariant is the dimension of the killed subspace of \(V_T(n)\), or equivalently the quotient rank of \(V_T(n)\).
Corollary 1 (Clique-counting generalized Turán theorem). For \(2\le s\le q+1\), \(\operatorname{ex}(n,K_s,K_{q+1})=t_s(n,q)\), where \(t_s(n,q)\) is the number of \(s\)-cliques in \(T_2(n,q)\).
Proof. Fix a squarefree monomial \(y_M\in C_{K_{q+1},n}\). We associate to \(M\) the vertex-variable quotient \[A_M= \Bbbk[z_1,\ldots,z_n]/(z_i^2,\;z_iz_j:y_{ij}\in M).\] Because \(y_M\) lies in the cover ideal of the \(K_{q+1}\)-copy ideal, every \((q+1)\)-vertex set \(L\subset[n]\) contains a pair \(ij\subset L\) with \(y_{ij}\in M\). Hence \(\prod_{i\in L}z_i=0\) in \(A_M\) for every \(|L|=q+1\), and therefore \((A_M)_{q+1}=0\). By Theorem 5, \(\dim_\Bbbk(A_M)_s\le t_s(n,q)\).
It remains to identify this Hilbert function with the quotient rank on the \(K_s\)-copy space. The linear map \[\Phi_s:V_{K_s}(n)\longrightarrow \Bbbk[z_1, \ldots,z_n]_s, \qquad \Phi_s\bigl(y_{E(K_W)}\bigr)=\prod_{i\in W}z_i\] identifies the monomial basis indexed by \(s\)-vertex complete subgraphs with the squarefree degree-\(s\) monomials in the vertex variables. After quotienting by \(\mathfrak m_M\) on the source and by the quadratic relations defining \(A_M\) on the target, the same basis elements are killed: \(y_{E(K_W)}\) is killed in \(R_n^{(2)}/\mathfrak m_M\) if and only if some \(y_{ij}\in M\) with \(i,j\in W\), and this is equivalent to \(\prod_{i\in W}z_i=0\) in \(A_M\). Thus \[\dim_\Bbbk\pi_M(V_{K_s}(n))=\dim_\Bbbk(A_M)_s\le t_s(n,q).\] Theorem 9 gives \(\operatorname{ex}(n,K_s,K_{q+1})\le t_s(n,q)\).
For equality, take a balanced partition of \([n]\) into \(q\) parts and let \(M_0\) be the set of all pairs contained in a single part. Then \(y_{M_0}\in C_{K_{q+1},n}\), and the quotient \(A_{M_0}\) is the complete balanced \(q\)-partite square-zero quotient. Its degree-\(s\) Hilbert function is \(t_s(n,q)\), so the upper bound is attained. ◻
Remark 11 (A quotient-rank cover-ideal strategy). Theorem 9 suggests the following method. Given a candidate extremal \(F\)-free object with missing-edge monomial \(y_{M_0}\), it suffices to prove \[\dim_\Bbbk\pi_M(V_T(n))\le \dim_\Bbbk\pi_{M_0}(V_T(n))\] for every squarefree monomial \(y_M\in C_{F,n}\). For \(T=K_2\) this is an initial-degree inequality. For \(T=K_s\) and \(F=K_{q+1}\), it is the Hilbert-function inequality of Theorem 5 applied to the vertex-variable quotient \(A_M\).
We now specialize the cover-ideal dictionary to the family \(\mathcal{K}_{\ell}^{(r)}\). For an \(r\)-graph \(G\subseteq\binom{[n]}r\), define its missing-edge monomial by \(m_G=\prod_{E\notin G}y_E\). Thus \(\deg m_G=\binom nr-|G|\).
For a pair \(\{a,b\}\subset[n]\), define the missing codegree-star monomial \[u^{(r)}_{ab}=\prod_{\substack{E\in\binom{[n]}r\\ \{a,b\}\subset E}}y_E.\] Then \(u^{(r)}_{ab}\mid m_G\) if and only if \(\operatorname{codeg}_G(a,b)=0\).
Definition 2 (Missing codegree-star ideal). For \(\ell,r\ge 2\), define \[J^{(r)}_{n,\ell}=\bigcap_{L\in\binom{[n]}{\ell}} \bigl(u^{(r)}_{ab}:\{a,b\}\subset L\bigr) \subset R_{n,r}.\]
Proposition 12 (Collapse of the cover ideal). For Mubayi’s family \(\mathcal{K}_{\ell}^{(r)}\), \(C_{\mathcal{K}_{\ell}^{(r)},n}=J^{(r)}_{n,\ell}\). Consequently, for every \(r\)-graph \(G\subseteq\binom{[n]}r\), \(m_G\in J^{(r)}_{n,\ell}\) if and only if \(G\) is \(\mathcal{K}_{\ell}^{(r)}\)-free.
Proof. Fix an \(\ell\)-set \(L\subset[n]\). Let \(C_L\) be the part of the cover ideal arising from forbidden copies with core \(L\), namely the intersection of the ideals \((y_E:E\in H)\) over all \(H\in \mathcal{K}_{\ell}^{(r)}\) whose core is \(L\). We claim that \(C_L=\bigl(u^{(r)}_{ab}:\{a,b\}\subset L\bigr)\). Suppose first that a monomial \(m\) is divisible by \(u^{(r)}_{ab}\) for some pair \(\{a,b\}\subset L\). Every \(H\in \mathcal{K}_{\ell}^{(r)}\) with core \(L\) contains an edge with the pair \(\{a,b\}\). The variable corresponding to that edge divides \(u^{(r)}_{ab}\), and hence divides \(m\). Thus \(m\in(y_E:E\in H)\) for every such \(H\), so \(m\in C_L\).
Conversely, suppose that \(m\) is not divisible by any \(u^{(r)}_{ab}\) with \(\{a,b\}\subset L\). For each pair \(\{a,b\}\subset L\), choose an \(r\)-edge \(E_{ab}\) containing \(\{a,b\}\) such that \(y_{E_{ab}}\nmid m\). The \(r\)-graph with edge set \(\{E_{ab}:\{a,b\}\subset L\}\) has core \(L\) and belongs to \(\mathcal{K}_{\ell}^{(r)}\). None of its edge variables divides \(m\), so \(m\) does not lie in the ideal generated by the variables of this forbidden copy. Hence \(m\notin C_L\).
The equality for each fixed core \(L\) follows. Intersecting over all \(L\in\binom{[n]}{\ell}\) gives \(C_{\mathcal{K}_{\ell}^{(r)},n}=J^{(r)}_{n,\ell}\). The final equivalence is the corresponding cover-ideal interpretation for the missing-edge monomial \(m_G\). ◻
Theorem 13 (Initial degree of the codegree-star ideal). For all \(n,\ell,r\ge 2\), \(\alpha\bigl(J^{(r)}_{n,\ell}\bigr)=\binom nr-t_r(n,\ell-1)\).
Proof. If \(n<\ell\), then the intersection defining \(J^{(r)}_{n,\ell}\) is empty, so \(J^{(r)}_{n,\ell}=R_{n,r}\). Also \(t_r(n,\ell-1)=\binom nr\), since the balanced \((\ell-1)\)-partite \(r\)-graph can place the \(n\) vertices in distinct parts. Thus both sides are zero. Hence assume \(n\ge \ell\).
Put \(q=\ell-1\). Since \(J^{(r)}_{n,\ell}\) is a squarefree monomial ideal, its initial degree is attained by a squarefree monomial. Let \(m\in J^{(r)}_{n,\ell}\) be squarefree. Define the associated vertex-variable quotient \[A_m= \Bbbk[x_1,\ldots,x_n]\Big/ \bigl(x_i^2,\;x_ax_b\text{ whenever }u^{(r)}_{ab}\mid m\bigr).\] This is a square-zero quadratic monomial quotient.
We first show that \((A_m)_\ell=0\). Let \(L\in\binom{[n]}{\ell}\). Since \(m\in \bigl(u^{(r)}_{ab}:\{a,b\}\subset L\bigr)\), there exists a pair \(\{a,b\}\subset L\) with \(u^{(r)}_{ab}\mid m\). By the definition of \(A_m\), this means \(x_ax_b=0\) in \(A_m\). Hence \(\prod_{i\in L}x_i=0\) in \(A_m\). Since this holds for every \(\ell\)-set \(L\), we have \((A_m)_\ell=(A_m)_{q+1}=0\). By Theorem 5, \(\dim_\Bbbk(A_m)_r\le t_r(n,q)=t_r(n,\ell-1)\).
Now consider an \(r\)-set \(E\in\binom{[n]}r\) such that \(y_E\nmid m\). We claim that \(x_E:=\prod_{i\in E}x_i\) is nonzero in \((A_m)_r\). If \(x_E=0\), then, since the defining ideal of \(A_m\) is generated by squares and squarefree quadratic monomials and \(E\) is squarefree, there is a pair \(\{a,b\}\subset E\) such that \(x_ax_b=0\) in \(A_m\). By construction of \(A_m\), this is equivalent to \(u^{(r)}_{ab}\mid m\). But \(E\) contains \(\{a,b\}\), so \(y_E\) is one of the factors in \(u^{(r)}_{ab}\). Hence \(y_E\mid m\), a contradiction.
Thus every variable \(y_E\) not dividing \(m\) gives a distinct nonzero standard monomial \(x_E\in(A_m)_r\). These monomials are linearly independent, so \[\#\{E\in\binom{[n]}r:y_E\nmid m\} \le \dim_\Bbbk(A_m)_r \le t_r(n,\ell-1).\] Because \(m\) is squarefree, \(\deg m=\binom nr-\#\{E\in\binom{[n]}r:y_E\nmid m\}\). Therefore \(\deg m\ge \binom nr-t_r(n,\ell-1)\). Since \(m\in J^{(r)}_{n,\ell}\) was arbitrary among squarefree monomials, this proves \(\alpha(J^{(r)}_{n,\ell})\ge \binom nr-t_r(n,\ell-1)\).
For the reverse inequality, take a balanced \((\ell-1)\)-partition \([n]=V_1\sqcup\cdots\sqcup V_{\ell-1}\). Let \[m_0=\prod_{\substack{E\in\binom{[n]}r\\ E\text{ contains two vertices from some }V_i}}y_E.\] Equivalently, \(m_0\) is the product of all non-transversal \(r\)-sets with respect to the partition. The transversal \(r\)-sets are exactly the edges of \(T_r(n,\ell-1)\), and hence \(\deg m_0=\binom nr-t_r(n,\ell-1)\). We verify that \(m_0\in J^{(r)}_{n,\ell}\). Let \(L\in\binom{[n]}{\ell}\). Since \(L\) has \(\ell\) vertices and there are only \(\ell-1\) parts, two vertices \(a,b\in L\) lie in the same part. Every \(r\)-set \(E\) containing \(\{a,b\}\) is non-transversal, so \(y_E\mid m_0\). Hence \(u^{(r)}_{ab}\mid m_0\), and therefore \(m_0\in \bigl(u^{(r)}_{ab}:\{a,b\}\subset L\bigr)\). Since \(L\) was arbitrary, \(m_0\in J^{(r)}_{n,\ell}\). Thus \(\alpha(J^{(r)}_{n,\ell})\le \deg m_0=\binom nr-t_r(n,\ell-1)\). Combining the two inequalities proves the theorem. ◻
Corollary 2 (Mubayi’s hypergraph Turán theorem via monomial ideals). For all \(n,\ell,r\ge 2\), \(\operatorname{ex}(n,\mathcal{K}_{\ell}^{(r)})=t_r(n,\ell-1)\).
Proof. By Theorem 7, Proposition 12, and Theorem 13, \[\begin{align} \binom nr-\operatorname{ex}(n,\mathcal{K}_{\ell}^{(r)}) &=\alpha(C_{\mathcal{K}_{\ell}^{(r)},n}) \\ &=\alpha(J^{(r)}_{n,\ell}) \\ &=\binom nr-t_r(n,\ell-1). \end{align}\] Canceling \(\binom nr\) gives the result. ◻
H.L. was supported by the National Natural Science Foundation of China (12501487), by China Scholarship Council and IBS-R029-C4. X.L. was supported by the Excellent Young Talents Program (Overseas) of the National Natural Science Foundation of China.
The authors used generative AI tools to assist in discussing proof strategies, checking proofs, and improving exposition. All mathematical arguments, results, and conclusions were reviewed and verified by the authors.
School of Mathematics, Shandong University, Jinan, China, and Extremal Combinatorics and Probability Group, Institute for Basic Science, Daejeon, South Korea. Email: heng.li@sdu.edu.cn.↩︎
School of Mathematical Sciences, University of Science and Technology of China, Hefei, China. Email: liuxizhi@ustc.edu.cn.↩︎