Is there an ultrafilter version of the “polynomial” Hales-Jewett theorem?

This theorem (as formulated in the paper of Walters you linked) is about (sequences of) multi-dimensional arrays with the same length at each dimension, instead of just one dimensional sequences, so there is no natural semigroup operation analogous to concatenation (that I can see).

That would also be a counterexample, but a rather degenerate one; the counterexample I selected was intended to resemble the standard Szemeredi example , to illustrate that non-degenerate averaging in r is not sufficient by itself to obtain recurrence.

]]>Wouldn’t also work, and look more natural? ]]>

In Lemma 3, Ellis-Nakamura lemma should be Ellis–Numakura lemma. see

http://en.wikipedia.org/wiki/Ellis-Numakura_lemma

*[Corrected, thanks – T.]*

I failed when doing exercise 11. Could you give one more hint, besides the fact that it is similar with the proof of Theorem 4 from the previous lecture. ]]>

Fair enough (though, in view of Lemma 1, one could also permit Q to vary with the if one wished.)

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