  We explore the properties of non-piecewise syndetic sets with positive upper
density, which we call "discordant", in countably infinite amenable
(semi)groups. Sets of this kind are involved in many questions of Ramsey theory
and manifest the difference in complexity between the classical van der
Waerden's theorem and Szemer\'{e}di's theorem. We generalize and unify old
constructions and obtain new results about these historically interesting sets.
Along the way, we draw from various corners of mathematics, including classical
Ramsey theory, ergodic theory, number theory, and topological and symbolic
dynamics.
