for a sequence a of integers, s(a) denotes the collection of partial sums of a. about forty years ago, erdos and folkman made the following conjecture: let a be an infinite sequence of integers with density at least $cn^{1/2}$ (i.e., a contains at least $cn^{1/2}$ numbers between $1$ and n for every larger n), then s(a) contains an infinite arithmetic progression. partial results have been obtained by erdos (1962), folkman (1966), hegyvari (2000), luczak-schoen (2000). together with szemeredi, we have recently proved the conjecture. in this talk, i plan to survey this development.