Avoiding arithmetic progressions in cyclic groups

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

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