ABSTRACT
In an earlier paper ([1]), the author and D. R. Farkas gave a new proof of G. Bergman’s theorem that there is no finitely generated algebra with GK dimension in the interval(1,2). Bergman deduces the result from the following proposition about words in the free monoid on finitely many letters: if B is a set of words closed under taking subwords and there is a positive integer d such that B contains at most d words of length d, then B has at most d 3 words of length h for each h ≥ d. Our proof, which essentially establishes the sharp bound of ( d + 1 2 ) 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315139975/1413b990-0cd9-4945-9840-4ee4de0fda23/content/eq502.tif"/> rather than d 3, depends on the description of a graph. We shall use the same construction to prove results about finitely presented monomial algebras.
