Abstract: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.
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.24507 [math.CO] |
| (or arXiv:2605.24507v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24507 arXiv-issued DOI via DataCite (pending registration) |
From: Heng Li [view email]
[v1]
Sat, 23 May 2026 10:40:42 UTC (14 KB)
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