Rothberger gaps in fragmented ideals
The Rothberger number (ℐ) of a definable ideal ℐ on ω is the least cardinal κ such that there exists a Rothberger gap of type (ω,κ) in the quotient algebra (ω)/ℐ. We investigate (ℐ) for a class of ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is ℵ₁, while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum...