  According to the Furstenberg-Zimmer structure theorem, every
measure-preserving system has a maximal distal factor, and is weak mixing
relative to that factor. Furstenberg and Katznelson used this structural
analysis of measure-preserving systems to provide a perspicuous proof of
Szemer\'edi's theorem. Beleznay and Foreman showed that, in general, the
transfinite construction of the maximal distal factor of a separable
measure-preserving system can extend arbitrarily far into the countable
ordinals. Here we show that the Furstenberg-Katznelson proof does not require
the full strength of the maximal distal factor, in the sense that the proof
only depends on a combinatorial weakening of its properties. We show that this
combinatorially weaker property obtains fairly low in the transfinite
construction, namely, by the $\omega^{\omega^\omega}$th level.
