How few three-term arithmetic progressions can a subset S ⊆ ZN := Z/NZ have if |S| ≥ υN? (that is, S has density at least υ). Varnavides [4] showed that this number of arithmetic-progressions is at least c(υ)N for sufficiently large integers N ; and, it is well-known that determining good lower bounds for c(υ) > 0 is at the same level of depth as Erdös’s… (More)