Colourful and Fractional (p, q)-theorems

Let p ≥ q ≥ d+1 be positive integers and let F be a finite family of convex sets in Rd. Assume that the elements of F are coloured with p colours. A p-element subset of F is heterochromatic if it contains exactly one element of each colour. The family F has the heterochromatic (p, q)-property if in every heterochromatic p-element subset there are at least q elements that have a point in common. We show that, under the heterochromatic (p, q)-condition, some colour class can be pierced by a finite set whose size we estimate from above in terms of d, p, and q. This is a colourful version of the famous (p, q)theorem. (We prove a colourful variant of the fractional Helly theorem along the way.) A fractional version of the same problem is when the (p, q)-condition holds for all but an α fraction of the p-tuples in F . We show that, in the case that d = 1, all but a β fraction of the elements of F can be pierced by p−q+1 points. Here β depends on α and p, q, and β → 0 as α goes to zero.