  Menger's basis property is a generalization of $\sigma$-compactness and
admits an elegant combinatorial interpretation. We introduce a general
combinatorial method to construct non $\sigma$-compact sets of reals with
Menger's property. Special instances of these constructions give known
counterexamples to conjectures of Menger and Hurewicz. We obtain the first
explicit solution to the Hurewicz 1927 problem, that was previously solved by
Chaber and Pol on a dichotomic basis.
  The constructed sets generate nontrivial subfields of the real line with
strong combinatorial properties, and most of our results can be stated in a
Ramsey-theoretic manner.
  Since we believe that this paper is of interest to a diverse mathematical
audience, we have made a special effort to make it self-contained and
accessible.
