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Buchstaber invariants of skeleta of a simplex

A moment-angle complex $\mathcal{Z}_K$ is a compact topological space associated with a finite simplicial complex $K$. It is realized as a subspace of a polydisk $(D^2)^m$, where $m$ is the number of vertices in $K$ and $D^2$ is the unit disk of the complex numbers $\C$, and the natural action of a torus $(S^1)^m$ on $(D^2)^m$ leaves $\mathcal{Z}_K$ invariant. The Buchstaber invariant $s(K)$ of $K$ is the maximum integer for which there is a subtorus of rank $s(K)$ acting on $\mathcal{Z}_K$ freely. The story above goes over the real numbers $\R$ in place of $\C$ and a real analogue of the Buchstaber invariant, denoted $s_\R(K)$, can be defined for $K$ and $s(K)\leqq s_\R(K)$. In this paper we will make some computations of $s_\R(K)$ when $K$ is a skeleton of a simplex. We take two approaches to find $s_\R(K)$ and the latter one turns out to be a problem of integer linear programming and of independent interest.