Small sets which meet all the k(n)-term arithmetic progressions in the interval [1, n]

For given n, k, the minimum cardinal of any subset B of [1, n] which meets all of the k-term arithmetic progressions contained in [1, n] is denoted by ƒ(n, k). We show, answering questions raised by Professor P. Erdos, that ƒ(n, nϵ) < C · n1−ϵ for some constant C (where C depends on ϵ), and that ƒ(n, log n) = o(n). We also discuss the behavior of ƒ(p2, p) when p is a prime, and we give a simple lower bound for the function associated with Szemeredi's theorem.