Szemeredi proved that every subset of the natural numbers of upper density epsilon > 0 contains arbitrarily long arithmetic progressions. If epsilon = 1, then this is obvious (to me). My question is whether there are any values of epsilon < 1 which allow Szemeredi's original elementary proof to be significantly simplified. Or is the argument equally hard for all positive epsilon? I'm struggling with the original elementary proof (as indicated in an earlier posting) and I'm wondering whether I can plug in the gap by arguing that we can assume epsilon < (for example) 0.9 by showing that we can quickly prove the result for large epsilon.