  In this paper we study generalizations of classical results on intersection
patterns of set systems in $\mathbb{R}^d$, such as the fractional Helly theorem
or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a
forbidden homological minor.
  Given a simplicial complex $K$ and an integer $b$, we say that a family
$\mathcal{F}$ of subcomplexes of some simplicial complex $\mathcal{U}$ is a
\emph{$(K,b)$-free cover} if (i) $K$ is a forbidden homological minor of
$\mathcal{U}$, and (ii) the $j$th reduced Betti number
$\tilde{\beta}_j(\bigcap_{S\in {\mathcal{G}}}S,\mathbb{Z}_2)$ is strictly less
than $b$ for all $0\leq j < \dim K$ and all nonempty subfamilies
$\mathcal{G}\subseteq \mathcal{F}$.
  We show that for every $K$ and $b$, the fractional Helly number of a
$(K,b)$-free cover is at most $\mu(K)+1$, where $\mu(K)$ is the maximum sum of
the dimensions of two disjoint faces in~$K$. This implies that the assertion of
the $(p,q)$-theorem holds for every $p \ge q > \mu(K)$ and every $(K,b)$-free
cover $\mathcal{F}$. For $b=1$ and a suitable $K$ this recovers the original
$(p,q)$-theorem and its generalization to good covers. Interestingly, our
results show that that the range of parameters $(p,q)$ for which the
$(p,q)$-theorem holds is independent of $b$.
  Our proofs use Ramsey-type arguments combined with the notion of stair
convexity of Bukh et al. to construct (forbidden) homological minors in cubical
complexes.
