Avoiding arithmetic progressions in cyclic groups

For given natural numbers n and r, α(n, r) denotes the maximum cardinality of a subset of Zn which does not contain any non-constant arithmetic progression (modulo n) of length r. The function α(n, r) is investigated for several values of n and r. In particular, it is shown that α(n, n) = n ( 1 − 1 p ) , where p is the smallest prime dividing n, and that for any prime number p we have α(p2, p) = (p− 1)2.