Bounding the Piercing Number

It is shown that for every k and every p ≥ q ≥ d + 1 there is a c = c(k, p, q, d) < ∞ such that the following holds. For every family H whose members are unions of at most k compact, convex sets in R in which any set of p members of the family contains a subset of cardinality q with a nonempty intersection there is a set of at most c points in R that intersects each member of H. It is also shown that for every p ≥ q ≥ d+ 1 there is a C = C(p, q, d) <∞ such that for every family G of compact convex sets in R so that among any p of them some q have a common hyperplane transversal, there is a set of at most C hyperplanes that together meet all the members of G.