  The Furstenberg-Zimmer structure theorem for $\mathbb{Z}^d$ actions says that
every measure-preserving system can be decomposed into a tower of primitive
extensions. Furstenberg and Katznelson used this analysis to prove the
multidimensional Szemer\'edi's theorem, and Bergelson and Liebman further
generalized to a polynomial Szemer\'edi's theorem. Beleznay and Foreman showed
that, in general, this tower can have any countable height. Here we show that
these proofs do not require the full height of this tower; we define a weaker
combinatorial property which is sufficient for these proofs, and show that it
always holds at fairly low levels in the transfinite construction
(specifically, $\omega^{\omega^{\omega^\omega}}$).
