Hilbert functions of monomial ideals containing a regular sequence

Let M be an ideal in K[x₁,..,x_n] (K is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing M satisfy the Eisenbud–Green–Harris conjecture and moreover prove that the Cohen–Macaulay property is preserved. We conclude that monomial ideals satisfy this conjecture. We obtain that the h-vector of Cohen–Macaulay simplicial complex Δ is the h-vector of Cohen–Macaulay (a₁ - 1,.., a_t - 1)-balanced simplicial complex, where t is the height of the Stanley–Reisner ideal of Δ and (a₁,.., a_t) is the type of some regular sequence contained in this ideal.