  We give a combinatorial characterization of amenability of monomial algebras
and prove the existence of monomial Folner sequences, answering a question due
to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization
to prove that over projectively simple monomial algebras, every module is
exhaustively amenable; we conclude that convolution algebras of minimal
subshifts admit the same property. We deduce that any minimal subshift of
positive entropy gives rise to a graded algebra which does not satisfy an
extension of Vershik's conjecture on amenable groups, proposed by Bartholdi.
Finally, we show that non-amenable monomial algebras must contain
noncommutative free subalgebras. Examples are given to emphasize the sharpness
and necessity of the assumptions in our results.
