Shelton–Yuzvinsky conjecture on Koszul Orlik–Solomon algebras

The conjecture asks whether Koszulness of an arrangement's Orlik–Solomon algebra forces the arrangement to be supersolvable. Earlier work established the implication in one direction and the converse only for several special classes.

Known results

• Supersolvability implies Koszulness for arbitrary matroids; the converse was conjectural (2012).
• The equivalence was proved for hypersolvable and graphic arrangements (2012).
• Root ideal arrangements satisfy the equivalence (2014).
• Cones over Dirichlet arrangements satisfy the equivalence (2018).

September 2026 counterexample claim

A preprint, Koszul Orlik–Solomon Algebras from Non-supersolvable Arrangements, claims three construction families of counterexamples, including arrangements of every rank at least three, and analogous examples for Orlik–Terao algebras. If correct, this disproves the conjecture in general; the constructions and their properties are unrefereed.

Current status (as of September 2026): The conjecture has a claimed general disproof via unrefereed counterexamples, while the counterexamples and their asserted Koszulness remain independently unverified.