The famous theorem of Szemer edi asserts that for every positive integer k and every positive real number > 0 there is a positive integer N such that every subset of f1; 2; : : :; Ng of cardinality at least N contains an arithmetic progression of length k. A second proof of the theorem was given by Furstenberg using ergodic theory, but neither this proof nor Szemer edi's gave anything other than extremely weak information about the dependence of N on k and. In this article we describe a new… CONTINUE READING