Geodesic
On the Hilbert Series of Koszul Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 2, pp. 64-69.

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A family of examples is obtained which shows that, generally, it is impossible to decide for known Hilbert series of a qudratic algebra and its dual algebra whether or not this algebra has the Koszul property. The simplest example is given by two finitely generated algebras concentrated at the degrees not exceeding five; one of these algebras is monomial, while the other is not a Koszul algebra. This proves the conjecture of Positselskii [pos]. 
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D. I. Piontkovskii. On the Hilbert Series of Koszul Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 2, pp. 64-69. http://geodesic.mathdoc.fr/item/FAA_2001_35_2_a5/

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