Lower bounds of tower type for Szemerédi's uniformity lemma

  title={Lower bounds of tower type for Szemer{\'e}di's uniformity lemma},
  author={W. T. Gowers},
  journal={Geometric \& Functional Analysis GAFA},
  • W. T. Gowers
  • Published 1997
  • Mathematics
  • Geometric & Functional Analysis GAFA
Abstract. It is known that the size of the partition obtained in Szemerédi'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… Expand
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