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We show that if a $d$-dimensional Cohen-Macaulay complex is, in a certainsense, sufficiently "close" to being balanced, then there is a $d$-dimensionalbalanced Cohen-Macaulay complex having the same $f$-vector. This in turnprovides some partial evidence for a conjecture of Kalai on the $f$-vectors ofCohen-Macaulay flag complexes.

Face Numbers of Certain Cohen-Macaulay Flag Complexes