  In this paper, we introduce the notion of a complete hypertetrahedral
arrangement $\mathcal{A}$ in $\mathbb{P}^{n}$. We address two basic problems.
First, we describe the local freeness of $\mathcal{A}$ in terms of smaller
complete hypertetrahedral arrangements and graph theory properties,
specializing the Musta\c{t}\u{a}-Schenck criterion. As an application, we
obtain that general complete hypertetrahedral arrangements are not locally
free. In the second part of this paper, we bound the initial degree of the
first syzygy module of the Jacobian ideal of $\mathcal{A}$.
