On piercing (pseudo)lines and boxes

We say a family of geometric objects C has (l;k)-property if every subfamily C0C of cardinality at most lisk- piercable. In this paper we investigate the existence of g(k;d)such that if any family of objects C in Rd has the (g(k;d);k)-property, then C is k-piercable. Danzer and Gr̈ unbaum showed that g(k;d)is infinite for fami-lies of boxes and translates of centrally symmetric convex hexagons. In this paper we show that any family of pseudo-lines(lines) with (k2+k+ 1;k)-property is k-piercable and extend this result to certain families of objects with discrete intersections. This is the first positive result for arbitrary k for a general family of objects. We also pose a relaxed ver-sion of the above question and show that any family of boxes in Rd with (k2d;k)-property is 2dk- piercable.