Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs

The collapsibility number of simplicial complexes was introduced by Wegner in order to understand the intersection patterns of convex sets. This number also plays an important role in a variety of Helly type results. There are only a few upper bounds for the collapsibility number of complexes available in literature. In general, it is difficult to establish such non-trivial upper bounds. In this article, we construct a sequence of upper bounds $\theta_k(X)$ for the collapsibility number of a simplicial complex $X$. We also show that the bound given by $\theta_k$ is tight if the underlying complex is $k$-vertex decomposable. We then give an upper bound for $\theta_k$ and therefore for the collapsibility number of the non-cover complex of a hypergraph in terms of its covering number