k-dimensional transversals for balls

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.