Maximal Betti numbers of Cohen–Macaulay complexes with a given f-vector

Given the f-vector f = (f0, f1, . . .) of a Cohen–Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Δf with f(Δf) = f such that, for any Cohen–Macaulay simplicial complex Δ with f(Δ) = f, one has \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\beta_{ij}(I_\Delta) \leq \beta_{ij}(I_{{\Delta}_{f}})$$ \end{document} for all i and j, where f(Δ) is the f-vector of Δ and where βij(IΔ) are graded Betti numbers of the Stanley–Reisner ideal IΔ of Δ.