  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.
