Intersection patterns in spaces with a forbidden homological minor
Xavier Goaoc, Andreas F. Holmsen, Zuzana Patáková
TL;DR
This work extends Helly-type intersection results to triangulable spaces with a forbidden homological minor by introducing $(K,b)$-free covers and proving that the fractional Helly number is at most $\mu(K)+1$, which implies a $(p,q)$-theorem for all $p\ge q>\mu(K)$ independent of $b$. The authors develop a robust framework based on grid complexes, stair-convex chains, and constrained chain maps to transfer intersection patterns from fixed-size subfamilies to larger ones, and to handle colorful (multipartite) configurations via a density-propagation mechanism. This leads to a unified topological approach that recovers classical results for appropriate $K$ (e.g., good covers) and provides a path toward a broader conjectural program on polynomial homological shatter functions and Leray-type nerves. The results have implications for topological combinatorics and potential algorithmic applications in property testing and constraint satisfaction by linking intersection patterns to global topological structure, with independence from the ambient dimension encoded in $\mu(K)$.
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 $\tildeβ_j(\bigcap_{S\in {\mathcal{G}}}S,\mathbb{Z}_2)$ 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 $μ(K)+1$, where $μ(K)$ 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 $p \ge q > μ(K)$ 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.