Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.24507 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917527822532608 |
|---|---|
| author | Li, Heng Liu, Xizhi |
| author_facet | Li, Heng Liu, Xizhi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24507 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Li, Heng Liu, Xizhi Combinatorics 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. |
| title | Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.24507 |