Let k > 3 be an integer, and let G be a finite abelian group with |G| = N , where (N, (k − 1)!) = 1. We write rk(G) for the largest cardinality |A| of a set A ⊆ G which does not contain k distinct elements in arithmetic progression. The famous theorem of Szemerédi essentially asserts that rk(Z/NZ) = ok(N). It is known, in fact, that the estimate rk(G) = ok… (More)