Description
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 R^d a constant fraction of the (k+2)-tuples of balls have a k-flat intersecting them, then there is a k-flat intersecting a constant fraction of all the balls. We show colorful, spherical, and infinite variants as well. Joint work with Dömötör Pálvölgyi.