Lower bounds of tower type for Szemeredi's Uniformity Lemma

It is known that the size of the partition obtained in Szemeredi's Uniformity Lemma can be bounded above by a number given by a tower of 2s of height proportional to epsilon(-5), where epsilon is the desired accuracy. In this paper, we first show that the bound is necessarily of tower type, obtaining a lower bound given by a tower of 2s of height proportional to log(1/epsilon). We then, give a different construction which improves the bound, even for certain weaker versions of the statement.