On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Tur\'{a}n Theorem

Heng Li; Xizhi Liu

REVIEW 1 major objections 1 minor 14 references

Mubayi's conjecture equating certain hypergraph ideals fails for all nontrivial parameters, though an alternative algebraic argument recovers the extremal formula for clique expansions.

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2026-06-30 13:25 UTC pith:FFCDASYJ

load-bearing objection Disproves the multipartite-ideal conjecture cleanly and supplies a distinct algebraic route to Mubayi's extremal formula, though the symmetrization step needs explicit hypothesis verification. the 1 major comments →

arxiv 2605.24507 v2 pith:FFCDASYJ submitted 2026-05-23 math.CO

On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Tur\'{a}n Theorem

Heng Li , Xizhi Liu This is my paper

classification math.CO

keywords Mubayi conjecturehypergraph Turán theorempolynomial idealsclique expansionsmonomial cover idealsHilbert functionsextremal combinatorics

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tests Mubayi's proposed algebraic analogue of Turán's theorem for hypergraphs, which conjectures that an ideal generated by multipartite 3-graphs equals a differentiated diagonal-vanishing ideal. Direct comparison shows the two ideals differ throughout the nontrivial parameter range. The authors nevertheless recover Mubayi's known extremal formula for the family of clique expansions by working instead with monomial cover ideals and invoking a Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients. A sympathetic reader cares because the disproof separates the original conjecture from the extremal statement it was intended to support, while the new argument shows that algebraic methods can still reach the same formula through different machinery.

Core claim

Mubayi's polynomial-ideal conjecture does not hold in the full nontrivial range of parameters. For the family K_ℓ^(r) of clique expansions, the extremal number is recovered algebraically by computing the Hilbert function of the quotient by the monomial cover ideal and applying the symmetrization theorem for square-zero quadratic monomial quotients.

What carries the argument

Monomial cover ideals of the clique-expansion hypergraphs, together with the Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients.

Load-bearing premise

The Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients applies directly to the monomial cover ideals arising from the clique-expansion family.

What would settle it

An explicit computation of the Hilbert function for the quotient by the monomial cover ideal of a small clique-expansion hypergraph that fails to match the symmetrized prediction would falsify the alternative proof.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The conjectured equality between the multipartite-3-graph ideal and the differentiated diagonal-vanishing ideal does not hold.
Mubayi's extremal formula for clique expansions admits an algebraic proof that bypasses the failed conjecture.
The symmetrization theorem produces the correct Hilbert function when applied to the relevant monomial quotients.

Where Pith is reading between the lines

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

Cover ideals may serve as a workable algebraic proxy in other hypergraph Turán problems where the original construction fails.
The disproof indicates that multipartite 3-graphs do not generate the ideal structure originally expected.
The same symmetrization technique could be checked on further families of hypergraphs to see whether the extremal formula extends algebraically.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Disproves the multipartite-ideal conjecture cleanly and supplies a distinct algebraic route to Mubayi's extremal formula, though the symmetrization step needs explicit hypothesis verification.

read the letter

The paper's main contributions are a disproof of Mubayi's conjecture on when multipartite 3-graph ideals coincide with differentiated diagonal-vanishing ideals, and an alternative algebraic proof of the extremal number for the clique-expansion family K_ℓ^(r).

The disproof holds across the nontrivial parameter range and stands on its own. The new proof route via monomial cover ideals plus the cited Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients is not in the prior literature referenced in the abstract, so that counts as genuine novelty.

The algebraic approach is a real strength: it separates the extremal result from the conjecture that has now been refuted, and it gives a different toolkit for the same formula.

The soft spot is narrow but real. The alternative proof invokes the symmetrization theorem directly on the cover ideals arising from K_ℓ^(r). The abstract states the application but does not record a check that those specific ideals satisfy the square-zero and quadratic conditions required by the theorem. If the full manuscript supplies that verification, the proof is fine; if not, the symmetrization step does not go through and the argument collapses. That is the only load-bearing uncertainty.

This is specialized work for people already inside hypergraph Turán theory or algebraic methods in extremal combinatorics. A reader looking for new proof techniques in that corner would find it useful. It is coherent on its own terms and shows clear engagement with the literature, so it deserves a serious referee even if the symmetrization check turns out to need more detail.

Referee Report

1 major / 1 minor

Summary. The paper disproves Mubayi's polynomial-ideal conjecture (that an ideal generated by multipartite 3-graphs coincides with a differentiated diagonal-vanishing ideal) throughout the nontrivial parameter range. It then supplies an alternative algebraic proof of Mubayi's extremal formula for the clique-expansion family K_ℓ^(r), relying on monomial cover ideals together with a Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients.

Significance. If the disproof stands and the alternative proof is valid, the work is significant: it both refutes a natural algebraic conjecture motivated by Li-Li ideals and supplies a new algebraic route to a known hypergraph Turán result, potentially opening further applications of cover ideals and symmetrization in extremal combinatorics.

major comments (1)

[Abstract and the section presenting the alternative algebraic proof] The alternative proof (invoked in the abstract and developed in the section on the algebraic proof of Mubayi's formula) applies the Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients directly to the monomial cover ideals constructed from K_ℓ^(r). The manuscript supplies no explicit verification that these specific cover ideals satisfy the square-zero and quadratic hypotheses of the theorem; without that check the symmetrization step is not justified and the proof does not go through.

minor comments (1)

[Abstract] The abstract is concise but omits any indication of the concrete parameter values or small cases used to establish the disproof; adding a brief sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the missing verification in the algebraic proof section. We address the comment below.

read point-by-point responses

Referee: [Abstract and the section presenting the alternative algebraic proof] The alternative proof (invoked in the abstract and developed in the section on the algebraic proof of Mubayi's formula) applies the Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients directly to the monomial cover ideals constructed from K_ℓ^(r). The manuscript supplies no explicit verification that these specific cover ideals satisfy the square-zero and quadratic hypotheses of the theorem; without that check the symmetrization step is not justified and the proof does not go through.

Authors: We agree that the manuscript does not contain an explicit verification that the monomial cover ideals arising from the clique expansions K_ℓ^(r) satisfy the square-zero and quadratic hypotheses of the cited Hilbert-function symmetrization theorem. This constitutes an omission in the exposition of the alternative proof. In the revised version we will add a short, self-contained paragraph (or subsection) that directly checks these two hypotheses for the specific cover ideals in question, thereby justifying the invocation of the symmetrization result. revision: yes

Circularity Check

0 steps flagged

No circularity: disproof and alternative proof are independent of the conjecture

full rationale

The paper states two distinct contributions: a direct disproof of Mubayi's conjecture across the nontrivial range, followed by a separate algebraic proof of the extremal formula that invokes monomial cover ideals together with a Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients. Neither step is shown to reduce to its own inputs by construction, nor does the text indicate that the symmetrization theorem is derived from the conjecture being disproved or from self-referential definitions within the paper. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard algebraic facts about polynomial rings, monomial ideals, and Hilbert functions; no free parameters, invented entities, or ad-hoc axioms are visible.

axioms (1)

standard math Standard properties of polynomial rings, monomial ideals, and Hilbert functions hold as background.
Invoked implicitly for all constructions involving ideals and quotients mentioned in the abstract.

pith-pipeline@v0.9.1-grok · 5664 in / 1215 out tokens · 33869 ms · 2026-06-30T13:25:33.522356+00:00 · methodology

0 comments

read the original abstract

Among the many proofs of Tur\'{a}n's classical theorem, one particularly surprising proof due to Li and Li uses ideals in polynomial rings to record missing edges. Motivated by their proof, Mubayi proposed a hypergraph analogue, conjecturing that an ideal generated by multipartite $3$-graphs coincides with a differentiated diagonal-vanishing ideal. If true, this conjecture would imply the extremal-number part of Mubayi's classical hypergraph extension of Tur\'{a}n's theorem. We disprove this conjecture throughout the nontrivial parameter range. We then give an alternative algebraic proof of Mubayi's extremal formula for the family $\mathcal{K}_{\ell}^{(r)}$ of clique expansions, using monomial cover ideals and a Hilbert-function symmetrization theorem for square-zero quadratic monomial quotients.

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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A. Kostochka, D. Mubayi, and J. Verstra¨ ete,Tur´ an problems and shadows III: expansions of graphs, SIAM J. Discrete Math. 29 (2015), no. 2, 868–876

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Mubayi and J

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[1] [1]

Alon and C

N. Alon and C. Shikhelman,ManyTcopies inH-free graphs, J. Combin. Theory Ser. B 121 (2016), 146–172

work page 2016

[2] [2]

J. A. Eagon and V. Reiner,Resolutions of Stanley–Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265–275

work page 1998

[3] [3]

Erd˝ os and A

P. Erd˝ os and A. H. Stone,On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946), 1087– 1091

work page 1946

[4] [4]

Kostochka, D

A. Kostochka, D. Mubayi, and J. Verstra¨ ete,Tur´ an problems and shadows III: expansions of graphs, SIAM J. Discrete Math. 29 (2015), no. 2, 868–876

work page 2015

[5] [5]

S.-Y. R. Li and W.-C. W. Li,Independence numbers of graphs and generators of ideals, Combinatorica 1 (1981), no. 1, 55–61

work page 1981

[6] [6]

Liu,New short proofs to some stability theorems, European J

X. Liu,New short proofs to some stability theorems, European J. Combin. 96 (2021), 103350

work page 2021

[7] [7]

Miller and B

E. Miller and B. Sturmfels,Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, Springer, New York, 2005

work page 2005

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Mubayi,A hypergraph extension of Tur´ an’s theorem, J

D. Mubayi,A hypergraph extension of Tur´ an’s theorem, J. Combin. Theory Ser. B 96 (2006), no. 1, 122–134

work page 2006

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Mubayi and J

D. Mubayi and J. Verstra¨ ete,A survey of Tur´ an problems for expansions, inRecent Trends in Combi- natorics, IMA Vol. Math. Appl., vol. 159, Springer, Cham, 2016, 117–143

work page 2016

[10] [10]

Pikhurko,Exact computation of the hypergraph Tur´ an function for expanded complete 2-graphs, J

O. Pikhurko,Exact computation of the hypergraph Tur´ an function for expanded complete 2-graphs, J. Combin. Theory Ser. B 103 (2013), no. 2, 220–225

work page 2013

[11] [11]

G. A. Reisner,Cohen–Macaulay quotients of polynomial rings, Adv. Math. 21 (1976), no. 1, 30–49

work page 1976

[12] [12]

R. P. Stanley,The upper bound conjecture and Cohen–Macaulay rings, Studies in Appl. Math. 54 (1975), no. 2, 135–142

work page 1975

[13] [13]

Tur´ an,Eine Extremalaufgabe aus der Graphentheorie, Mat

P. Tur´ an,Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452

work page 1941

[14] [14]

A. A. Zykov,On some properties of linear complexes, Mat. Sb. (N.S.) 24(66) (1949), 163–188. 11

work page 1949