A stepping-up lemma for topological set systems

Intersection patterns of convex sets in $\mathbb{R}^d$ have the remarkable property that for $d+1 \le k \le \ell$, in any sufficiently large family of convex sets in $\mathbb{R}^d$, if a constant fraction of the $k$-element subfamilies have nonempty intersection, then a constant fraction of the $\ell$-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system $\mathcal{F}$ in $\mathbb{R}^d$. Quantitatively, our bounds depend on how complicated the intersection of $\ell$ elements of $\mathcal{F}$ can be, as measured by the sum of the $\lceil\frac{d}2\rceil$ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to $d+1$. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matousek and Nivash to recast a simplicial complex as a homological minor of a cubical complex.