Abstract
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. Erdös, that ƒ(n, n ϵ) < C · n 1−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(p 2, p) when p is a prime, and we give a simple lower bound for the function associated with Szemerédi's theorem.
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