We prove fractional Helly and $(p,k+2)$-theorems for $k$-flats intersecting Euclidean balls. For example, we show that if for a collection of balls from $\mathbb R^d$ any $p$ balls have a $k$-flat that intersects at least $k+2$ of them, then the whole collection can be intersected by bounded many $k$-flats. We prove colorful, spherical, and infinite variants as well. In fact, we prove that fractional Helly and $(p,q)$-theorems imply $(\aleph_0,q)$-theorems in an entirely abstract setting. The fractional Helly theorems generalize to other fat objects as well.

翻译：我们证明了与欧几里得球相交的k-平面的分数Helly定理和$(p,k+2)$-定理。例如，我们证明：若来自$\mathbb R^d$的一组球中任意$p$个球存在一个k-平面与其中至少$k+2$个球相交，则整个集合可以被有界多个k-平面相交。我们还证明了彩色、球面及无穷变体。事实上，我们证明在完全抽象的框架下，分数Helly定理和$(p,q)$-定理可推出$(\aleph_0,q)$-定理。这些分数Helly定理也可推广至其他胖体对象。