Some of the combinatorics related to Michael's problem

We present some new methods for constructing a Michael space, a regular Lindelöf space which has a non-Lindelöf product with the space of irrationals. The central result is a combinatorial statement about the irrationals which is a necessary and sufficient condition for the existence of a certain class of Michael spaces. We also show that there are Michael spaces assuming $\mathfrak{d}= \operatorname{cov}(\mathcal{M})$ and that it is consistent with $\operatorname{cov}(\mathcal{M}) < \mathfrak{b} < \mathfrak{d}$ that there is a Michael space. The influence of Cohen reals on Michael's problem is discussed as well. Finally, we present an example of a Michael space of weight less than $\mathfrak{b}$ under the assumption that $\mathfrak{b} = \mathfrak{d}= \operatorname{cov} (\mathcal{M}) = \aleph _{\omega +1}$ (whose product with the irrationals is necessarily linearly Lindelöf).