Stanley s Conjecture on the k-Cohen-Macaulay simplicial complexes of codimension 3

Let $Delta$ be a simplicial complex on vertex set $[n]$. ##It is shown that if $Delta$ is k-Cohen--Macaulay of codimension 3, then $Delta$ is vertex decomposable.## As a consequence we show that $Delta$ is partitionable and Stanley s conjecture holds for $K[Delta]$.