Let I be a homogeneous ideal of a polynomial ring over a field, v(I) the number of elements of any minimal basis of I, e = e(I) the multiplicity or degree of R/I, h = h(I) the height or codimension of I, i = indeg (I) the initial degree of J, i.e. the minimal degree of non zero elements of I.
This paper is mainly devoted to find bounds for v(I) when I ranges over large classes of ideals. For instance we get bounds when I ranges over the set of perfect ideals with preassigned codimension and multiplicity and when I ranges over the set of perfect ideals with preassigned codimension, multiplicity and initial degree. Moreover all the bounds are sharp since they are attained by suitable ideals. Now let us make some historical remarks.