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, Matou\v{s}ek and Nivash to recast a simplicial complex as a homological minor of a cubical complex.

翻译：$mathbb{R ⁇ d$ $\\ mathb{R ⁇ d$ 的 convex 的交叉模式具有显著的属性, $+1\le k\le k\ ell$, 任何足够大的 convex 的家族, $\ mathb{R ⁇ d$, 如果$$- eleconfamilies 的固定部分是非空的交叉点, 那么$@ ell- ele- eleffamilies 的固定部分也必须有非空的交叉点 。 在此, 我们证明对于任何 $\ mathcal{ F} 的复杂结构, $\ mathb{ R ⁇ d$ 。 从数量上看, 我们的界限取决于 $ellexliformx 元素的交叉点是如何复杂的 。 我们用$lasseyC 的 comlicrecial 格式向 $Dal+1xrequeral 提供一些约束性精度的系统。