ABSTRACTLet S=K[x1,…,xn] be a polynomial ring over a field K and I be a nonzero graded ideal of S. Then, for t≫0, the Betti number βq(S∕It) is a polynomial in t, which is denoted by 𝔅qI(t). It is proved that 𝔅qI(t) is vanished or of degree l(I)−1 provided I is a monomial ideal generated in a single degree or grade(𝔪R(I)) = codim(𝔪R(I)) where 𝔪=(x1,…,xn) and R(I) is the Rees ring of I. One lower bound for the leading coefficient of 𝔅qI(t) is given. When I is a Borel principal monomial ideal, 𝔅qI(t) is calculated explicitly.