We study parameters of the convexity spaces associated with families of sets in $\mathbb{R}^d$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$. Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems.

翻译：我们研究$\mathbb{R}^d$中集合族所关联的凸性空间参数，其中族内任意$t$个集合的交集的Betti数均被$t$的某个函数从上界控制。尽管此类集合族的Radon数可能无界，但我们证明这些族满足分数Helly定理。为此，我们引入了Radon数和Helly数的分级类比。这一结论推广了先前已知的分数Helly定理。