On Mubayi's Polynomial-Ideal Conjecture and a Hypergraph Turán Theorem

Abstract

Among the many proofs of Turán'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 Turán'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.