Abstract
Theorem 1 is due to Folkman [4], who also asked whether its assertion remains true if e > 0 is replaced by a function which tends to 0 as n→∞. Theorem 2 below states that this is indeed the case and, furthermore, for every set A dense enough, one can take b = 0. It should be mentioned that recently a similar result has been independently proved by Hegyvari [5], who showed that the assertion of Theorem 1 holds for all A ⊆ N with A(n) > 300 √ n log n for n large enough.
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