Intersection patterns in spaces with a forbidden homological minor (2103.09286v3)

Abstract: 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.