On the history of van der Waerden’s theorem on arithmetic progressions
- Tom C. Brown, Jau-Shyong Shiue
For given r-z, k, the minimum cardinal of any subset B of [l, n] which meets all of the k-term arithmetic progressions contained in Cl, n] is denoted by f(n, k). We show, answering questions raised by Professor P. Erdiis, that f(n, ne) < C . n’-’ for some constant C (where C depends on E), and that f(n, log n) = o(n). We also discuss the behavior of f(p2, p) when p is a prime, and we give a simple lower bound for the function associated with Szemeredi’s theorem.