Is there some minimal idempotent ultrafilter $q \in \beta( \mathbb{N}^2)$ (with respect to the law $"+"$) such that any $A \in q$ is a subset of $\mathbb{N} \times \{ 0 \} $ ? (See for example http://www.math.osu.edu/~bergelson.1/vbkatsiveli20march03.pdf for definitions).

Motivation : Van der Waerden theorem can be quickly deduced from a negative answer to this question.