  We present two new problems on lower bounds for resolution Betti numbers of
monomial ideals generated in a fixed degree. The first concerns any such ideal
and bounds the total Betti numbers, while the second concerns ideals that are
quadratic and bihomogeneous with respect to two variable sets, but gives a more
finely graded lower bound.
  These problems are solved for certain classes of ideals that generalize (in
two different directions) the edge ideals of threshold graphs and Ferrers
graphs. In the process, we produce particularly simple cellular linear
resolutions for strongly stable and squarefree strongly stable ideals generated
in a fixed degree, and combinatorial interpretations for the Betti numbers of
other classes of ideals, all of which are independent of the coefficient field.
