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arXiv · 2605.24507

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

Heng Li·Xizhi Liu
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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.

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Heng Li, Xizhi Liu. 2026-05-23. On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Tur\'{a}n Theorem. https://arxiv.org/abs/2605.24507

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