A universal bound for surfaces in 3-manifolds with a given Heegaard   genus
  It is shown that for given positive integers g and b, there is a number C(g,b), such that any orientable compact irreducible 3-manifold of Heegaard genus g has at most C(g,b) disjoint, nonparallel incompressible surfaces with first Betti number b_1 < b. 
Mario Eudave-Munoz and Jeremy Shor