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Let me first recall some facts:

  1. By the work of Gowers, the Van der Waerden numbers belong to class $\mathcal{E}^3$ of the Grzegorczyk hierarchy

  2. By the work of Shelah, the Hales-Jewett numbers belong to class $\mathcal{E}^5$.

But there exists a higher dimensional version of the Van der Waerden theorem known as the Gallai-Witt theorem. As it follows from the Hales-Jewett theorem, so the corresponding numbers belong to class $\mathcal{E}^5$ as well.

Question. Is it known if the Gallai-Witt numbers belong to the class $\mathcal{E}^3$ or even $\mathcal{E}^4$?

If so, may you please provide a reference.

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