Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem

Noga Alon, Daniel J Kleitman
1992 Advances in Mathematics  
A family of sets has the (p, q) property if among any p members of the family some q have a nonempty intersection. It is shown that for every p > q > d + 1 there is a c = c(p, q, d) < cc such that for every family % of compact, convex sets in Rd which has the (p, q) property there is a set of at most c points in Rd that intersects each member of 9. This settles an old problem of Hadwiger and Debrunner.
doi:10.1016/0001-8708(92)90052-m fatcat:autlyssvpnhtflbg6x43whjeia