Van der Waerden’s Theorem is false for $latex \omega_1 $

One of my main research problems involves something I think is related to arithmetic progressions in \mathbb{Z} . After learning a nice proof of Van der Waerden’s Theorem, and sharing it with colleagues, Daniel Soukup asked a nice question about Van der Waerden’s Theorem on \omega_1 . We answered it, and the example was sufficiently nice that I would like to share it.

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