  Given a homogeneous ideal $I \subseteq k[x_0,\dots,x_n]$, the Containment
problem studies the relation between symbolic and regular powers of $I$, that
is, it asks for which pair $m, r \in \mathbb{N}$, $I^{(m)} \subseteq I^r$
holds. In the last years, several conjectures have been posed on this problem,
creating an active area of current interests and ongoing investigations. In
this paper, we investigated the Stable Harbourne Conjecture and the Stable
Harbourne -- Huneke Conjecture and we show that they hold for the defining
ideal of a Complement of a Steiner configuration of points in
$\mathbb{P}^{n}_{k}$. We can also show that the ideal of a Complement of a
Steiner Configuration of points has expected resurgence, that is, its
resurgence is strictly less than its big height, and it also satisfies
Chudnovsky and Demailly's Conjectures. Moreover, given a hypergraph $H$, we
also study the relation between its colourability and the failure of the
containment problem for the cover ideal associated to $H$. We apply these
results in the case that $H$ is a Steiner System.
